ASVAB: MATH

The tricks to find some of the prime numbers are: - Any even number is divisible by 2. - If you add up the digits in a large number and the sum you get is divisible by __________, the number is also divisible by 3. - A number that ends with a 5 or 0 is divisible by ___________.

2, 3, 5

If the terms you are working with have _________________ __________________, there is not much you can do to simplify the expression. The only exception to this rule is if both the bases are __________________________. Then, to simplify, you can simplify each ___________ and then ________________. Simplify 4^3 / 2^3. 4^3 = __________ and 2^3 = ___________. You can simplify further and then divide. __________/__________ = ___________.

different, bases, numbers, term, divide, 64, 8, 64, 8, 8

Solve: y = -3x2 + 15x + 29. We substitute in y = 0 and begin thinking about which way we want to solve. We will solve this using the quadratic formula. This means identifying a, b and c from our standard form quadratic. That makes a equal to __________, b equal to __________ and c equal to __________. Now it's just a matter of carefully substituting these values into the quadratic formula and then following the order of operations to come up with our two answers. Substituting the numbers in will look like this: x= ___________ plus or minus (√ ____________- 4 (________) (__________)) /2 (________). In the formula, it's a -b, so in this case 15 turns into -15. But often times your b value itself will be negative. When you have a -b and you plug it into the formula, - -b will turn it ___________________________ again, so you've got to be careful with that b in front. Anyway, moving on to evaluating, we can start with the exponent on the inside - the b2 part of the discriminant - and 152 gives us ________________. Next, we do 4 * __________ * __________ to get ______________, and we do 2 * __________ to get ___________ on the bottom. Again, changing a minus negative into plus a positive will make our discriminant equal to _________________. The square root of 573 is about 23.937. Doing -15 plus this and dividing by -6 will give us -1.489, and doing -15 minus this divided by -6 will give us 6.489.

-3, 15, 29, -15, 152, -3, 29, -3, positive, 225, -3, 29, -348, -3, -6, 573,

Factor 2x2 - 5x - 3. First, we need to find a pair of numbers that will add up to the middle coefficient - in this case, __________. But the second part is going to be different. Instead of our two numbers needing to have a product equal to the constant on the end, we now need the product of our pair of numbers to be equal to the constant on the ____________ times the _______________________ coefficient. In this case, our end term is ____________ and our leading coefficient is __________, so the product is __________. This was actually true for the easier factoring problems as well, but the leading coefficient was just 1, so multiplying by 1 didn't change what the number was. The next step of the problem should be familiar. Find a pair of numbers that has a sum of -5 and a product of -6. _________ and _________ are our two winners. Instead of simply being able to say -6 and +1 go in our binomials and we're done, there is an extra step before we can be sure of our answer. While there are multiple ways of doing this step, I recommend using the area method to work your way backward to the answer. Putting the 2x^2 and the -3 in one diagonal of the chart and the two values we came up with using our pattern attached with x's in the other diagonal, gets us one step away from our answer. If we can find what terms must have been on the outside of this chart to get multiplied in and give us what we have here, we'll be done. We do this by dividing out the ________________________ ____________________ _______________________ from each row and column of our chart. So, if I look at the top row of this chart, I have a 2x2 and -6x. I need to ask myself what do those things have in common? Well, 2 and 6 are both divisible by ____________, so I can take out a _________. But they also both have an x, which means I also take out an ___________, so I can pull a ___________ to the outside of that row. Going down a row to the bottom, 1x and -3, they don't have any factors with the numbers in common, and they also don't have any variables in common, which means the only thing I can divide out is a __________. Now, we go to the columns. Let's start in the left column, 2x2 and 1x. The numbers don't have anything in common but the variables do, which means I can take out __________. Finally, the column on the right: -6x and -3. Both share a __________, which means I divide that out and write it on top. What we now have on the left and above our little area model is our factored answer. The terms that are on the same side are the terms that go in parentheses together to make up our two binomials, and I end up with (2x + 1)(x - 3).

-5, end, leading, -3, 2, -6, -6, 1, greatest, common, factor, 2, 2, x, 2x, 1, 1x, -3,

Quadratic Formula= x= (______________ plus or minus √ _________^__________ -________________ ) / ___________________________________________ Standard Form ___________= _____________^________+______________+____________

-b, b^2, 4ac, 2(a), y, ax, 2, bx, c

What if we were asked to simplify the square root of 2 times the square root of the quantity 200 divided by 3. If a is greater than ________ and b is greater than _______, then we can use the following properties to simplify square root expressions: First, we have the quotient property, where the square root of a / b equals the square root of ______ divided by the square root of _______. We also have the product property, where the square root of a*b is the same as the square root of ______ times the square root of ______.

0, 0, a, b, a, b,

Let's say that, in this example, it was shot up into the air and came back down and travelled 2 feet in its first trip. Then, when it bounced and hit the ground, it travelled 1/4 of the distance. So if it went up and down 2 feet the first time, the second time it would only go up and down half a foot. Then, it would only go up and down .125 feet and then it would continually bounce smaller and smaller amounts. It would go, theoretically, forever, to infinity. ' The rule for this sequence would be an = ________ * (._________)^(n - ________). 2 is the ________________ ________________ and the _________________ ____________________ is 1/4. We want to take the sum, from the first through the infinite term of that rule, and now I can just use my formula: (________)/(________ - _________). a1 is _________ and r is ______/_______. My formula turns into _________/(1 - _______/_________). 1 - 1/4 is _______/_______. Dividing by a fraction is the same thing as multiplying by its reciprocal. I get 2 * 4/3. Putting a 1 under the 2 and multiplying across gives me 8/3, or around 2.6667 feet.

2, .25, 1, 2, 1/4, beginning value, common ration, a1, 1, r, 2, 1/4, 2, 1/4, 3/4

What if we have exponents? Here's one: y^3 + 9y^2. What happens when you multiply terms with exponents? So y^2 is just y * y. And y^3 is y * y * y. If we look at it this way, each term has ________ y's in it, or y^________. Let's factor out a y^________. If you take a y^2 from y^3, you're left with just one ________. And if you take a y^2 from 9y^2, you get just _________. So, our factored expression is y^_________(_________+ _________). Okay, let's graduate to something more complex: p^3q^2 + pq^3. Both terms have a p. And both terms also have a q. More than that, they both have a q^2, just like in that last example. So, we can factor out __________^________. If we take pq^2 out of p^3q^2, the q^2 factors to _________ and the p^3 factors to p^_________. With pq^3, the p factors to _________ and the q^3 factors to just __________. That makes our factored expression pq^_________(p^_________ + __________).

2, 2, 2, y, 9, 2, y, 9, pq, 2, 1, 2, 1, q, 2, 2, q

In math, we define a quadratic equation as an equation of degree ___________, meaning that the highest exponent of this function is ___________. The standard form of a quadratic is y = ___________^__________ + _____________ + ____________, where a, b, and c are numbers and a cannot be ___________. An interesting thing about quadratic equations is that they can have up to __________ real solutions. Solutions are where the quadratic equals __________. Real solutions mean that these solutions are not imaginary and are ________________ numbers. Imaginary numbers are those numbers with an imaginary part: __________. We will solve the quadratic 2x^2 + 4x = 0. Next, we look for our greatest common factor. We see that it is a ___________. We go ahead and factor that out to get ___________(_________ + _________) = ___________. Now, to find our solutions, we set both factors equal to 0 like this. ___________ = 0 and _________ + __________ = 0. Solving both for x, we get x = ___________ and x = _____________

2, 2, ax, 2, bx, c, 0, 2, 0, real, i, 2x, 2x, x, 2, 0, 2x, x, 2, 0, -2

We can use this information to find the least common multiple by following three steps. First, complete the prime factorization for each number. We just did that for 36 which was 3 x 3 x 2 x 2. Let's do it for 40. 40 is __________ x __________), 2 is prime. So, _________ is _________ x _________ and 10 is _________ x _________, so 40's prime factors are __________ x __________ x __________ x ___________. Now we're done with step one. Second, find which prime number occurs most often. So both sets have 2s, but there are more 2s with 40; that's what I mean by 'occurring most often.' Then there are two 3s with 36 and one 5 with 40. In these cases, that's most often because they don't occur in the other sets of factors. Third, find the product of these numbers. Okay, we're almost there. This will be cool. So, 2 x 2 x 2 x 3 x 3 x 5. What is that? Well 2 x 2 is 4, 4 x 2 is 8, 8 x 3 is 24, 24 x 3 is 72, and 72 x 5 is 360. Guess what? 360 is the least common multiple of 36 and 40. We can make sure it's a multiple by dividing each number into it. 360 / 36 is 10. 360 / 40 is 9.

2, 20, 20, 2, 10, 2, 5, 2, 2, 2, 5,

Any number written in scientific notation has two parts. The first part is the digits - written with a decimal point after the first number and excluding any leading or trailing zeros. Leading zeros are zeros between the decimal point and the first non-zero number in a number smaller than 1. For example, the number 0.0012 has _______ leading zeros. Trailing zeros are zeros after the last non-zero number in a number greater than 1. For example, the number 5,308,000 has _______ trailing zeros. The zero between the 3 and the 8 is not a trailing zero because there is a non-zero number (the ________) that comes after it. The second part of a scientific notation number is the x _______ to a power. This part puts the decimal point where it should be. It shows how many places to move the decimal point. 1.) Large numbers will have a __________________ exponent. 2.) Small numbers will have a ___________________ exponent.

2, 3, 8, 10, positive, negative

We decided that the rule for nth term was _______ * ________^(_______ - ______), because 2 was the _____________ term and the ____________________ ____________was 3; we kept on _________________________ by 3 each time. This means that if we want to know how many total views we got in the first two weeks, we're taking the ______________ of that exact rule from day 1 through day 14. Now I can just plug numbers into the formula. a1 is ________. In this case, n is __________ because we're going through 14 days. r is _________. I get ________((1 - ______^________)/(1 - ________)). This simplifies down into 2((1 - 4,782,969)/(1 - 3)). Keeping on and subtracting, I get 2(-4,782,969/-2). Doing the division, negative divided by negative is a positive. Then, doing the multiplying, and it looks like our video got 4,782,968 views in the first two weeks alone.

2, 3, n, 1, first, common ratio, multiplying, sum, 2, 14, 3, 2, 3^14, 3

2, 6, 18, 54, 162, 486 We started at a1=__________, then went up to a2=_________ (2x3), then a3=_________ (2x3x3), then 54 happened because we multiplied by 3 again (2x3x3x3). I keep on multiplying by 3, so the next one is 2x3x3x3x3. We can start condensing these multiplications by 3 into exponents, which means the sixth term would be 2x3^_______. Notice that the sixth term had an exponent of ________ on the 3. If I condense the 3s into exponents on the fifth term, I get the _______th power; if I condense them on the fourth term, I get the _______rd power; the third term's the _______nd; the second term's the _______st; and the first terterm-6m we can call the 0 power, because anything to the 0 power is __________, so we just get 2 x 1, or 2. Generalizing the pattern for any term, this means that for any day n after I posted the video, the number of new hits is 2x3^n-1. The 2 in front represents the beginning value _________, the 3 represents what would be called the '_________________ _______________' and the n-1 represents how many times we had to multiply by 3. It's n-1 because we didn't get any new hits until the ______________________ day. Now to figure out how many new hits I'm going to get after two weeks is pretty straightforward. I want to know how many new hits I'm going to get on the 14th day, so we substitute ____________ in for n. That gives us that a________=2x3^(______-______) power. We do 3^13 power and we get 1,594,323. We multiply that by 2 and it turns out that 14 short days after I posted the video I'm getting 3,188,646 new hits!

2, 6, 18, 5, 5, 4, 3, 2, 1, 1, a1, common ratio, second, 14, 14, 14-1,

We can use the quotient property to help with the square root of the quantity 200 divided by 3 part of our problem. Following this property, we can rewrite this part as the square root of ______________ divided by the square root of ________. We know from earlier that the square root of 200 can be simplified to _______ times the square root of ________. Now, we have the square root of _______ times ______ times the square root of ________ in the numerator and the square root of ________ in the denominator. To simplify the numerator, we can use the product property to write the square root of _______ times the square root of _______ as the square root of the quantity _______ times ________. This gives us the square root of ________, so we now have 10 times the square root of _______ in the numerator. Don't forget that 4 is a perfect square, so we can rewrite the square root of 4 as ________. This gives us _________ in our numerator. square root example 2 still going strong Are we done? No, remember that there can't be a square root sign in the denominator, so how do we get rid of that square root of 3? We have to rationalize the denominator. This is a fancy way of saying that we need to multiply by some version of 1 to get rid of square root signs in the denominator. In this case, we multiply our entire expression by the square root of ________ divided by the square root of _________. Multiplying our denominator by the square root of 3 gives us the square root of 3 times 3 (remember the product property). This gives us the square root of ________ in our denominator, which is just 3. Our final, simplified version is ________ times the square root of ________ all over ________. The radicands have no perfect square factor other than 1 and there are no square root signs in the denominator, so we're done!

200, 3, 10, 2, 2, 10, 2, 3, 2, 2, 2, 2, 4, 4, 2, 20, 3, 3, 9, 20, 3, 3

Okay, 18a^3 + 9a^2. Let's do this. If we just wanted to factor numbers, we'd look at that 18 and 9 and find the greatest common factor. With 18 and 9, it's _________. If we just wanted to factor variables, we'd look at that a^3 and a^2. We can factor out an a^________. So, _________^________ is the greatest common factor of the terms in this expression. If we pull out 9a^2 from 18a^3, what are we left with? Well, the 18 becomes a ________, and the a^3 becomes just _________, since a^2 * a is a^________. The factored expression is _________^_________(________) Let's try one with two variables and three terms: 24pq^2 + 32p^3q + 8p^2q. Again, start with the numbers. 24, 32 and 8. _________ is the greatest common factor of each number. What about the variables? We can take these separately. There's a p, a p^3 and a p^2. So, we can factor out a ___________. With the q's, there's a q^2 and then two q's. So, we'll factor out a ________. That's an 8, a p and a q. Put it together, and we have _________. If we factor 8pq from 24pq^2, we have ____________. If we factor 8pq from 32p^3q, we have ___________^_________. Let's check that. Finally, if we factor 8pq from 8p^2q, we're left with just ___________. So, our factored expression is _____________(___________ + ___________^________ + ____________).

9, 2, 9a, 2, 2, a, 3, 9a, 2, 2a, 8, p, q, 8pq, 3q, 4p, 2, p, 8pq, 3q, 4p, 2, p

We can also come up with the rule to a geometric sequence by simply being given any two entries. Say we know that a3=9/4 and a6=243/32. We want to know what the rule is for the nth term. I don't know what the first term or the second term is, but I do know that the third term is _______/________. The fourth term and the fifth term are unknown to me, but the sixth term is ___________/_________. Because it is a geometric sequence, I know that when I started with my third term (9/4) I simply multiplied by r ____________________ times in a row to get my sixth term, which means that 9/4r^_______ = __________/__________. Now what I've set up is an equation that I can solve using inverse operations to figure out what the common ratio r must be. I undo multiplication by a fraction by multiplying by its reciprocal, __________/___________, which gives me ________/________ on the right side. I can then undo a third power with a third root. I take the cubed root of both sides and find that the cubed root of 27 is _________ and the cubed root of 8 is __________, and I find that our common ratio is ______/_______. The only other thing needed for our rule besides r is a1, which does mean that I have to work my way backward to find a1. I can come back to my picture and divide by r. I divide once by 3/2 and get 3/2, and I divide one more time by 3/2 and find that a1 is simply equal to 1. This means that my rule an=a1r^n-1 is an = 1 x 3/2^n-1. Because multiplying by 1 doesn't actually change anything, I can just condense my rule to an=3/2^n-1.

9/4, 243/32, three, 3, 243/32, 4/0, 27/8, 3, 2, 3/2,

The FOIL Method Let's start with the most well known method: ______________. This is the method that uses an acronym to help you remember, and it stands for '_______________, ____________________, __________________, ________________'. Once we've done the multiplying, we again combine _________________ ________________ in the end to get our answer The Area Method Instead of having you remember an acronym like FOIL, the area method asks you to draw a ________________. Because each binomial has two terms, we'll draw a box for the problem (2x + 3)(3x - 1) with ______________ sections on each side. This gives us a box with ____________________ regions, one for each mini multiplication problem that we're about to do. Labeling each side with one of the _______________________ and giving each term its own section lays out all the mini multiplication we need to do. Now we just need to come up with what goes on the inside of the four regions by looking across and above to see which terms are labeled on the outside and multiplying those two things together to get our four mini multiplication problems. Once all four regions have been filled, we can rewrite the expression outside the chart, 6x^2 + 9x -3, and combine like terms to get our final answer: 6x^2 + 7x - 3.

FOIL, firsts, outsides, insides, lasts, like terms, chart, two, four, binomials,

P - _____________________________: Every operation in _________________________ should be performed first. E - ___________________________: All ________________________ should be calculated next. M - _________________________________ D - ___________________________: All multiplication and division should be done next, in order from _____________ to _____________. Not all multiplication, then all division, but both of them as they appear from left to right. A - ________________________ S - ___________________________: Finally, all addition and subtraction should be done in order from ______________ to ____________.

Parenthesis, parenthesis, Exponent, exponents, Multiplication, Division, left, right, Addition, Subtraction, left, right

We are going to talk about five exponent properties. Just like the order of operations, you need to memorize these operations to be successful. The five exponent properties are: Product of ___________________ Power to a ___________________ Quotient of __________________ Power of a ___________________ Power of a ____________________

Powers, Power, Powers Product, Quotient

Vertex Form And finally we come to vertex form: y = __________(__________ - ___________)__________ + ______________. This time, getting your quadratic into this form requires you to complete the square, which is possibly the hardest algebraic trick of them all. But if you can, you are going to be rewarded for your hard work. First off, the _________ value still tells us whether it's ________________________ up or down, and the y-____________________ is still easily found by substituting in __________= 0 and evaluating. But now, just like intercept form gave us the intercepts, vertex form will give us the ___________________ of our parabola straight from the equation: __________ is going to become the x-coordinate, and __________ will become the y-coordinate, of our vertex. Now, we can easily tell where the axis of symmetry is simply by remembering that it goes right through the _____________________ of the graph where the vertex is. Therefore, the axis of symmetry is just the line __________ = _________.

a, x, h, 2, k, a, concave, intercept, x, vertex, h, k, middle, x, h

Intercept Form The next form we'll go over is intercept form, y = _________(_________ - _________)(__________ - _________). The a value will, again, tell you whether the parabola is concave up or down, and if you want to find the y-intercept, you can simply substitute in x=0 and quickly evaluate __________(_________)(________). Where intercept form gets its name and passes standard form in usefulness, is in its ability to not just tell you where the y-intercept is but also where the __________-_______________________ are. Because the x-intercepts are where _________=0, substituting in either _________ or _________ will give you a _________________ in your product, turning the entire equation into ________________. Therefore, ___________ and _________ are the two x-intercepts, or roots, of your quadratic. Be careful with the signs on your roots, though. Because the general equation has a -p and -q, an (x - 5) would actually mean a root at x=__________, while an (x + 5) would mean a root at x= __________. Lastly, because parabolas are symmetrical, the axis of symmetry must lie directly in between the ______________ roots. This means you can find it on your graph by working your way into the middle or algebraically by calculating the average between the two points: x = (__________ + _________)/_________.

a, x, p, x, q, a, -p, -q, x, intercepts, y, p, q, zero, zero, p, q, 5, -5, two, p, q, 2

If we're taking the sum, from the first term to the nth term, of the geometric sequence (a1)(r^(n - 1)), that is simply equal to (______)((_____ - ______^_______)/(_______ - _______)). What we now have is our formula.

a1, 1, r^n, 1, r,

The first set of words include the words is, are, will be, gives, and similar words that show one thing matching up or being identified with another thing. When you see one of these words, you will write the ___________________ symbol (_________) for _______________________. The next set of words include the words more, total, sum, plus, combine, increased, and similar words that show things are being added together. When you see one of these words, write the ___________________ symbol (_______) for _______________________. Another set includes the words decreased, minus, less, fewer, takes, and similar words that show things are being taken away. Write the __________________ symbol (_______) for ____________________ when you see these words. The next set includes the words of, multiplied, product, times, and similar words that show things are being made larger by _______________________________. Use the __________________________ symbol (_______) for these words. The last group of words includes the words per, out of, ratio of, and similar words that show things are divided. Use either the ______________________ symbol or the _________________ slash (__________) for these words.

equals, =, equality, plus, +, addition, minus, -, subtraction, multiplication, multiplication, x, division, division, %

It turns out that this trick will work for any arithmetic series. Simply pair up the ____________ and the _______________ term (__________ + __________) and multiply by however many _________________ you have (______/______), and you get your sum! We, therefore, have the following formula written in summation notation. We start with the ___________ term and go to ___________, where n is the _____________ _______ ____________ in our series, and we're doing this for some arithmetic sequence an. What we get is (______/_______), which is the number of terms divided by ____________, times the sum of that ______________ and ________________ term (________ + _________). So, that gives us (_____/______)(_______ + ______), which is our formula for any arithmetic series.

first, last, a1, an, groups, n/2, first, n, number of terms, n/2, 2, first, last, a1, an, n/2, a1, an

-x^2 + 5x +6 < 0 The first part of the solution is to move everything to the ________________ side. We will use algebra skills to do this. After using our algebra skills, on the left, we have ____________^_________ + ___________ + ____________. To finish solving our quadratic inequality, we need to solve the combined quadratic for zero by temporarily changing the inequality sign into an equal sign. We can use what we know about factoring quadratics to find our zeroes. Our quadratic factors into this: (___________ - ___________) (___________ - ___________) = 0 After factoring, to find our zeroes, we set each factor equal to __________________, and solve for the variable. We set the first factor, -x - 1, equal to __________________ to find the first zero. Solving for x, we get __________ = ___________. We then multiply by __________ on both sides to get x = ___________ for our first zero. We set our second factor, __________ - __________, equal to zero to find the second zero. We then get x = _________ as our next zero. Our zeroes are located where x is ___________ and ____________. From the zeroes that we have just found, we will now label a number of ranges based on the number of zeroes. In our case, we have a range where x < -1, another range where x is > -1 and x < 6, and a third where x > 6.

left, -x, 2, 5x, 6, -x, 1, x, 6, zero, zero, -x, 1, -1, -1, x, 6, 6, -1, 6

When adding and subtracting terms that have variables, you may remember that you're supposed to combine _______________ _____________. We apply this same concept to adding and subtracting square roots, except we combine terms that have the same ______________________. Simplify the following radical: 6 times the square root of the quantity 1 over 2 plus 4 times the square root of 18 minus 8 times the square root of 2. That may look like a lot, but let's work on simplifying each term on its own. We can use the quotient property on our first term. This gives us _______ times the square root of ______ all over the square root of _______. The square root of 1 is just _______, so this is really _______ over the square root of _______. To finish simplifying this term, we need to rationalize the _________________________ to get rid of the _______________ ________________ sign. Multiply this first term by the square root of ________ over the square root of ________, and you'll see that this term simplifies to_______ times the square root of _______. Now, on to the second term. We can use the product property to help us find a perfect square. 4 times the square root of 18 becomes _______ times the square root of _______ times the square root of the perfect square 9. The square root of 9 is _______, so this gives us _______ times 3 times the square root of ________, or 12 times the square root of 2. Okay, let's look at our final term, negative 8 times the square root of 2. Is there anything we can do to simplify it further? No, the radicand has no perfect squares, and there's certainly no square root sign in the denominator. Our final step is to add and subtract these square roots. Remember that we can only combine terms that have the same ____________________, but all of our terms have the radicand _________. This means we can combine all of them and rewrite our expression as the square root of ________ times the quantity ________ + ________ - _________. This just gives us ________ times the square root of 2, and we're finished!

like terms, radicand, 6, 1, 2, 1, 6, 2, denominator, square root, 2, 2, 3, 2, 4, 2, 3, 4, 2, radicand, 2, 2, 3, 12, 8, 7

Here is an expression: 4x - 8. Let's say we want to factor it. We can define factoring as finding the terms that are _______________________ together to get an expression. Our expression here has some important parts. First, we have two terms: 4x and 8. The terms are the numbers, variables or numbers and variables that are multiplied together. Terms are separated by plus or minus signs. _________ is just a constant, or a number that is what it is. It's constantly 8. ________ is a variable, or a symbol standing in for a ________________________ we don't know. _________ is a coefficient. Notice the prefix 'co-.' The coefficient multiplies a ____________________. It's a codependent, cooperating coefficient.

multiplied, 8, x, number, 4, variable,

So we can assume from this that to simplify a term with a power raised to a power, you just need to multiply the exponents together. But let's try another example. Simplify: x^3 ^5 to simplify, you will multiply the exponents together _________ x __________= ____________ x^____________

multiply, 3, 5, 15, 15

The formula for Arithmetic Sequence an= dn + a0 an= ___________________ dn= ___________________ a0=____________________ All arithmetic rules will look something like this and they will be an = ________+_________. The a0 obviously represents the ___________________ value, kind of like the b or the y-intercept, and d(what used to be called the slope) is now going to be called the ________________ _________________ because each term in our sequence has a common difference between them. The Arithmetic Sequence formula can also be written as an= d (n-1) + a1 This allows us not to worry about converting the a1 to a0 in the original equation.

nth term, common difference, beginning value, dn, a0, beginning, common difference,

Write this term using only positive exponents. Simplify where possible. (x^-8 * 3^3) / (4^-2 * y^2) The first thing we can do is switch every term with a negative exponent to the ____________________ side of the ____________________________. x^-8 will move to the _____________________________, and 4^-2 will move to the ______________________________. Then our term looks like this: (_______^________ * ________^__________) / (_________^_________ * _________^_________) To simplify, we can multiply out the numbers with exponents. 4^2 becomes __________, and 3^3 is ________. Finally, we can multiply 27 and 16 to get: _____________ / (x^8 * y^2)

opposite, fraction, denominator, numerator, 4, 2, 3, 3, x, 8, y, 2, 16, 27, 432

Anytime you graph a quadratic equation you end up with what is called a ______________________. The maximum is the highest _________-value that the parabola reaches. The name of the actual point on the parabola where it gets to the maximum is our second vocab word; it's called the ___________________. You might say that the vertex is in the ____________________ of the parabola. That's because the parabolas are symmetrical, they're the same on either side. This means our third vocab word is the line that goes straight down through the middle of the parabola to divide it in half. It's called, the ________________ __________ ______________________. Just like any other graph, parabolas' intercepts where the curve _________________________ either the x or the y-axis. In parabola word problems, the x-intercepts will often be the place where your flying object hits the ground, just like here. These x-intercepts of quadratic equations (and also bigger functions) can also be called ____________________.

parabola, y, parabola, vertex, middle, axis, of, symmetry, intersects, roots

A sequence is just a set of things (usually numbers) that make a _____________________________. The two most common sequences are called arithmetic and geometric sequences, and in fact all the examples we've looked at so far fall into either one of these two categories. Arithmetic sequences are patterns formed with repeated _________________________. Our example 1, 2, 3, 4, 5... and on and on is an example of an arithmetic sequence because we were continually adding ______ to get each new term. Geometric sequences are two sequences that are formed with repeated __________________________. So 5, 10, 20, 40, 80... and on an on, was geometric because we simply multiplied by _________ to find each next term. The Fibonacci Sequence. One of the most famous sequences is called the Fibonacci sequence. This sequence is found by adding the __________________________ two entries to come up with the next one.

pattern, addition, 1, multiplication, 2, previous

A square root expression is considered simplified once it meets two conditions: 1) The radicands have no ___________________ square factors other than _________ and 2) There are no ____________________ ________________ signs in the ___________________________ What our first condition really means is we're looking for ______________________ squares under the square root sign. We can pull those perfect squares out from under the square root sign and write them as _______________________. For example, to simplify the square root of 200, you'd want to find the perfect square factors of 200. We can write 200 as ___________*100. Is ________ a perfect square? No, its principal square root is not an ___________________. Is 100 a perfect square? Yes, its ______________________ square root is 10! Let's pull that 100 out from under the radical sign and write it as __________. So, the square root of 200 can be simplified to __________ times the square root of _________.

perfect, 1, square root, denominator, perfect, integers, 2, integer, principal, 10, 10, 2

To factor a number is to break that number down into smaller parts. To find the prime factorization of a number, you need to break that number down to its _____________________ factors. Both methods start out with a factor tree. A factor tree is a diagram that is used to break down a number into its __________________________ until all the numbers left are _________________. Since 24 is an even number, the first prime number that can be factored out is a _________. This leaves us with ____________ * ___________. Again, ____________ is an even number, so we can factor out another ____________, leaving us with 2 * 2 * 6. Since 6 is even, we can factor out a third two, leaving ___________ * __________ * ____________ * ____________. All of these numbers are __________________, so the factorization is complete. The other method for using a factor tree to find the prime factorization of a number is just to pull out the first ______________________ that you see, whether they are prime or not. Looking back at our example from above, let's factor 24 again using this method. The first thing you might notice is that ____________ * 4 is 24, so that is one set of factors for 24. Since neither of these numbers are prime, we can continue to factor both of them. 6 can be broken down to ___________ * __________, and 4 can be broken down to ____________ * __________. Now all of our factors are prime, and the factorization of 24 is complete, again giving the answer of ____________ * ____________ * _____________ * _____________.

prime, factors, prime, 2, 2, 12, 12, 2, 2, 2, 2, 3, prime, factors, 6, 2, 3, 2, 2, 2, 2, 2, 3

What is prime factorization? This is when we're finding the _______________ numbers that ____________________ together to make a number. Let's unpack that a bit. The factors of a number are the numbers that, if you multiply together, you get the original number. Prime factors are factors that are ________________ ____________________, or numbers larger than one that can only be evenly divided by one or themselves.

prime, multiply, prime, numbers

When you have two exponential expressions that have the ____________ _____________, you can easily divide one from another. All you have to do in this instance is ___________________ the exponent of the denominator (the bottom number of a fraction) from the exponent of the numerator (the top number of a fraction). Here's an example: 5^7 / 5^2. To simplify this expression, just subtract the exponents: ________ - ________ = _________. So, the answer is ________^________. This simplification works with all exponential expressions where the base is the same for each term. If the base is _______________________, no simplification can be done. The rule also applies if one or more of the exponents are negative. Simplify b^-2 / b^6. Again, just subtract the exponents, making sure to subtract them in the proper order. ________ - _________ = _________. So, ________^_________/_________^_________ = ________^_______.

same, base, subtract, 7, 2, 5, 5, 5, different, -2, 6, -8, b, -2, b, 6, -8

This is where the Greek letter sigma comes in. Anytime you see this letter in math, it's implying that we'll be taking a ________________, which means that we're just ____________________ up the terms of a _______________________. What goes underneath the letter will tell you where we _______________ the sequence. So, it tells you the ________________________ term, n = something. The number on the top is where we _____________ the series. The only thing missing is the rule, and that goes directly to the _________________ of the sigma. What we have after we fill all those things in is either called sigma notation or summation notation.

series, adding, sequence, start, starting, end, right,

The least common multiple of two or more numbers refers to the _____________________ __________________ number that is divisible by those numbers. So we're not looking for the least common multiple as in the most rarely occurring. Rather, we want the least common multiple, as in the smallest __________________ ______________________. Let's say we have 3 and 5. To find the least common multiple, we just start listing the multiples of each. The multiples of 3 are 6, 9, 12, 15, 18, 21, 24, 27, 30, and so on. The multiples of 5 are 10, 15, 20, 25, 30, and so on. What multiples are shared? Well, of the ones we listed, there's ______________ and _____________. What's the smallest? ______________. So ______________ is the least common multiple of 3 and 5.

smallest, whole, shared, multiple, 15, 30, 15, 15,

Square root signs are telling us to find the ________________ _________________ of whatever numbers are beneath them, and the numbers beneath a square root sign is called a _______________________. As for square roots, a number r is a square root of a number x if r^2 = ________. Based on this definition, a positive number actually has two square roots: a __________________________ number and the __________________________ of that positive number. For example, the numbers 5 and -5 are square roots of _____________ because 5^2 = ____________ and (-5)^2 = _____________. You can think of every positive root as having a negative, evil twin. ______________________ squares provide us with a way to find the simpler versions of square root expressions. A radicand is a ____________________ square if its ________________________ square root is an ____________________. For example, 16 is a perfect square because its principal square root is ____________. The number 26 is not a perfect square because its principal square root (about 5.1) is not an _____________________.

square roots, radicand, x, positive, negative, 25, 25, 25, Perfect, perfect, principal, integer, 4, integer

-x^2 + 5x +6 < 0 What do we know about quadratic expressions? We know that, in general, a quadratic has three terms in it: a term with a squared variable, a term with a single variable, and a term with only a number. The ____________________________ variable term is the part that makes the expression a quadratic. We also know that a quadratic expression can also have less than three terms. As long as it has the _________________________ variable term, then it is a quadratic. Because we have two sides to our inequality, our original problem can have terms on both sides. As long as you end up with a quadratic after moving all your terms to one side, you are working with a quadratic inequality.

squared, squared

These three main forms that we graph parabolas from are called _____________________________ form, __________________________ form and __________________ form. Each form will give you slightly different information and have its own unique advantages and disadvantages. Standard Form Let's begin with standard form, y = ___________^________ + ____________ + __________. To be completely honest, the main reason this one makes the cut as a useful form is because it's the easiest and most basic to write. Once we get past that, though, standard form doesn't have too much to offer. Perhaps, its most useful trait is that the a value tells you whether the parabola is concave up (_______________________ ________ value) or concave down (______________________ ________ value), but it turns out that all the forms are going to have this ability. The second trait of standard form has to do with the y-intercept of the parabola. Since the y-intercept is where x=0, substituting this in shows us that the a and b terms drop out, leaving us with only the c value. Therefore, the c value is always the ________-_________________________. The last thing you can do with standard form is calculate the axis of symmetry with the formula __________ = ____________ / ___________.

standard, intercept, vertex, ax, 2, bx, c, positive, a, negative, a, c, y, intercept, x, -b, 2a

-2x (x - 1) + 5 + x = 0 Both ways we know how to solve quadratics require that the equation is in standard form (y = ___________^________ + ___________+ __________), and this one definitely is not. All we need to do is use our algebra skills to move things around and put this equation back into standard form. Let's start by distributing the -2x to the (x - 1) on the inside of the parentheses in order to change this expression only to addition. Doing that gives us this: ___________^________ + __________+ ___________+ __________= 0, and now it's only a matter of combining like terms to end up with our standard form quadratic, ___________^_________ + _____________ + _____________ = 0. We can now start thinking about solving this, and these numbers aren't too bad, so let's try factoring. This means first factoring out the negative from everything in the trinomial to give us this: -(____________^________ - ___________ - _________) = 0, and then finding a pair of numbers that have a product of -10 and a sum of -3. Writing out the factors of -10 and looking for the pair that fit our criteria makes it look like ____________ and ___________ are our winners. That means we can use the area method to factor this out. Substituting those values into the four quadrants and then taking the greatest common factor out of each column and row gives us the factored form of the equation as -(__________ - __________)(_________ + __________) = 0. That changes the equation in our problem to this: 0 = 2x - 5 or 0 = x + 1, which now allows us to use the zero product property. Now it's only a few quick inverse operations to find that our two answers are _________/__________ or _____________.

y, ax, 2, bx, c, -2x, 2, 2x, 5, x, -2x, 2, 3x, 5, -2x, 2, 3x, 5, -5, 2, 2x, 5, x, 1, 5/2, -1

Solve the following equation using the quadratic formula: y= -x^2 + 4x +7 You'll have to first remember the standard form of a quadratic equation, ___________ = ___________^________ + ___________+ ___________. This will tell you what your a, b and c values are, but then it simply becomes a matter of putting the numbers into the right spots. Going back to the original equation and identifying a, b and c should be our first step. A is in front of the _________^________, where there's only a negative symbol. That means that a is just __________. B is the number in front of the ___________, which makes it ___________, and c is the constant on the ____________, ____________. Plugging this into the quadratic equation gives us, x= ((_________ plus or minus √ _________^2 - 4(________)(________) )/ 2(_______) For this problem, then, the first thing we'll have to take care of is the exponent we see. The exponent I see is a ___________, which means a square, which means multiply that number by itself, so we do 4 * 4 to get ____________. Next in the order of operations will come multiplication. Doing the multiplication on the inside of the square root (__________ * __________ * _________) gives us ____________, then doing the multiplication on the bottom of the fraction (_________ * _________) is simply ___________. We're now left with x = (-4 +/- √(16 - (-28))) / -2, and we're going to focus on the inside of the square root in the top of the fraction. The inside of the square root in the formula is b2 - 4ac. This part is one of the hardest parts to evaluate and one of the easiest parts to make a mistake on. It also has a special name; it's called the discriminant. Minus a negative is plus a _____________________, so ___________ + __________ gives us ___________. The square root of 44 is in between __________ and _________; it's actually about 6.633. That means we'll find our two answers by doing -4 + 6.633/-2 and also -4 - 6.633/-2. Doing that gives us our two estimated answers as -1.316 or 5.316.

y, ax, 2, bx, c, x^2, -1, x, 4, end, 7, -4, 4, -1, 7, -1, 2, 16, 4, -1, 7, -28, 2, -1, -2, positive, 16, 28, 44, 6, 7,

The multiplication property of zero tells us that anything multiplied by zero will equal _______________. So, that tells us that if either of our factors equals 0, then that will be an answer. So to find what our x equals, we can set both of our factors, our x and x + 1, equal to __________ to solve for our x to find our answers. So, we get _________ = _________ and _________ + _________ = __________. Right away, we see that one solution is 0. To find our other solution we need to solve for x by subtracting ________ from both sides. When we do that we find our other solution is x = _________. At this point we are done! Let's look at another example to see how this method works. Another Example We will solve the quadratic 2x^2 + 4x = 0. The first thing that we notice is that we don't have a constant term, so we can use the method we just learned. Next, we look for our ______________________ common factor. We see that it is a _____________. We go ahead and factor that out to get ___________(_________ + _________) = __________. Now, to find our solutions, we set both factors equal to 0 like this. ___________ = __________ and __________ + __________ = __________. Solving both for x, we get x = _________ and x = ___________, and we are done!

zero, 0, x, 0, x, 1, 0, 1, -1, greatest, 2x, 2x, x, 2, 0, 2x, 0, x, 2, 0, 0, -2,

How about one more? -8a^2b - 4ab^2 - 16ab. Let's start with the numbers. We have 8, 4 and 16. Our greatest common factor is ___________. That's not all. We have -8, -4 and -16. They're all negative! So we can factor out that _______________________ sign, too. That means that our greatest common factor for the numbers isn't 4, but __________. They all have an a, but only one term has an a^2. And they all have a b, but only one term has a b^2. So, we can factor out an __________. Put that together, and we're factoring out a ____________. With the first term, we're left with a positive 2a. With the second term, we're left with positive _________. And, finally, the third term is just going to be positive ___________. So, our factored expression is _____________(__________ + __________ + ____________).

4, negative, -4. ab, -4ab, 2a, b, 4, -4ab, 2a, b, 4

It works with variables, too. Let's say we have this: 25a^4b^2c^6/ 20a^3b^2c^4 Let's simplify. We can think of this as four separate fractions, one for each variable and one for the constants, 25 and 20 - those are like the Brady parents. At least they're not wearing hats. Let's start with them, 25 and 20. Those simplify to _______ and ________. With the variables, we can just subtract the smaller exponent from the larger one. With a4 over a3, we're left with _______ over ________. No exponents! Awesome. Then there's b2 over b2. Well, those just go away, don't they? The middle child always seems to get left out. Okay, now c6 over c4. We simplify that to _______ over 1. If we put this all together, we have this: ______________/___________.

5, 4, a, 1, c2, 1, 5ac2/4

5y^3 - 3y^2 + 10y - 6 Step 1: Group it as (___________^_________ - ___________^________) + (____________ - ___________). Step 2: What can we factor from (5y^3 - 3y^2)? Well, nothing from 5 and 3, but we can factor out a ________. Not just that, but ________^________. We get _________^________(_________ - _________). What about (10y - 6)? We can't factor out a y, but 10 and 6 have a common factor in __________. We get _________(____________ - __________). Hey, look at that: 5y - 3 again! We found some common parts that go together. That leads us to Step 3: factor out that ___________ - __________. So we have (___________^_________ + ____________)(___________ - __________).

5y, 3, 3y, 2, 10y, 6, y, y, 2, y, 2, 5y, 3, 2, 2, 5y, 3, 5y, 3, y, 2, 2, 5y, 3

3x^2 + 2x + 12x + 8 With grouping, we break up the expression into smaller groups that can be factored. The first step is to group the first _______________ terms and the last _______________ terms. Think of these as separate sets of expressions. Here, let's do (3x^2 + 2x) and have that plus (12x + 8). We basically have two two-term expressions, or binomials. The second step is to factor the greatest common factor from each binomial. With (__________^_________ + ____________), we can factor out an x to get _________(___________ + _________). With (__________ + __________), we can factor out a 4 to get __________(___________ + __________). Okay, now we have ________(___________ + ____________) + _________(___________ + ____________). That leads to our final step. The third and final step is to factor out the common binomial. Do you see how we have two (3x + 2)'s? We can factor that out. That gets us (_________ + __________)(___________ + ___________).

two, two, 3x, 2, 2x, 12x, 8, x, 3x, 2, 4, 3x, 2, x, 3x, 2, 4, 3x, 2, x, 4, 3x, 2

Product of Powers Here's the formula: (_____^______)(______^_______) = x^(______ + _______). When you multiply exponentials with the _____________ base (notice that x and x are the same base), ______________ their exponents (or powers). Let me show you how that works. Let's say I have (x^2)(x^3). Well, x^2 is x times x, and x^3 is x times x times x. When we add all those x's up, we get x^_______, which is the same thing as adding _______ + ________.

x, a, x, b, a, b, same, add , 5, 3, 2

Power of a Quotient When we look at this formula, we have ______ /______, or a fraction raised to the a power, this gives us (x^______) / (y^______). When you have a quotient to a power, you give each base its own exponent. We think of it as the __________________ being distributed to each part of the fraction, just like the last one, power of a product. Let me show you how this one works. Let's say I have (x / y)^3. Remember, that means I'm going to take x / y and multiply it _______________ times. That would be x / y times x / y times x / y. If we look at the top (or the numerator), we have x times x times x, or x^_______. If we look at the bottom (or denominator), we have y times y times y, or y^________. That would give us (x^3) / (y^3), which is basically distributing _________ to the x and y .

x, y, a, a, exponent, three, 3, 3, 3

Let's look at the two formulas. x^-a = 1/(x^a) 1/(x^-a) = x^a If the exponential is negative in the denominator, or the bottom, it tells us the exponential is actually _________________________ in the _________________________. If you see a negative exponent, flip it to a _________________________. That is, if the exponent is negative in the numerator, flip it ____________________ to the ________________________. If the exponent is negative in the denominator, flip it ____________________ to the _________________________. Let's say I have x^-4. That's the same thing as saying __________/(x^_________).

positive, numerator, positive, positive, denominator, positive, numerator, 1, 4,

Two binomials are conjugates if they have the ________________ two _________________ but _______________________ signs on the second one like (a _______ b) and (a ______ b). Any time we multiply two conjugates together, the two ____________________ terms will drop out, just like we saw, giving us the shortcut: (a + b)(a - b) = _________^_________ - __________^__________

same, terms, opposite, +, -, middle, a, 2, b, 2

It's also good to be able to read something in sigma notation and understand what it's asking you to do. Say that you're asked to evaluate the following sum. I notice that underneath the sigma, I have n = 3, which means the first term in my series, the first term in the sequence that I'm going to be adding up, is the _____________ term, _______. The top is ___________, which means I'm going to stop after I get to the ___________term. All I have to do is __________ + ___________ + ____________. I know that the rule is n^2 - 1. So, a3 is just _______^2 - ________. a4 is __________^2 - _________. a5 is ________^2 - _________. I evaluate that out and 9 - 1 = ________, 4^2 - 1 = _________, 5^2 - 1 = __________. I add those numbers up: 8 + 15 + 24. And, it turns out that this sum would be equal to ____________.

third, a3, 5, fifth, a3, a4, a5, 3, 1, 4, 1, 5, 1, 8, 15, 24, 47

The rule for zero as an exponent is that any number or variable (except _______________ itself) raised to the 0 power is equal to ________. x^4 * x^0 = x^(4+0) = _______^_______ As you can see, nothing changed. So we need to ask ourselves the question, what is the only number that when you multiply something by it, nothing changes? The only answer to that question is _________. When you multiply something by 1, it does not change. So the only thing to conclude with our example above is that x^0 = ________ x^4 * x^0 = x^4 x^4 * 1 = x^4 Therefore, x^0 = _________ This pattern plays out no matter what numbers or variables we use, so we can say that any term (except ______________) raised to the zero power is equal to _____________.

zero, 1, x, 4, 1, 1, 1, zero, one

The formula we came up with works for all finite geometric series. But, as it turns out, it's also going to work for some infinite geometric series as well. If r, the common ratio, is bigger than ________, then the series just gets bigger and bigger and bigger, and it's never going to stop. Finding a specific sum is going to be impossible because it's just going to become _____________________. If, instead, r is in between _________ and _________, the sum is going to converge to a specific number. This is because the numbers that we're going to be adding on get so small that they basically turn into ________ and don't really do anything. So, our sum, from 1 to infinity, of our geometric series doesn't have the same equation, (a1)((1 - r^n)/(1 - r)). Now, that r^n in the old formula might be (1/2)^100, because we're going to infinity, so the n is going to get huge. And (1/2)^100 is like 0.00000... I don't even know; there's a ton of 0's there. This means that if we kept going, (1/2)^200, (1/2)^300, it just gets smaller and smaller. So, 1/2 to the infinity is going to turn into _________, which means that this _______^________ disappears from our formula and our formula turns in (a1)((1 - 0)/(1 - r)). I can then multiply the a1 to the 1 and my formula simplifies down to (_______)/(______ - ______).

1, infinity, 0, 1, 0, 0, r^n, a1, 1, r

Let's look closely at that last problem we just did. We said we could factor x2 + 8x + 15 as (x + ________)(x + _________). What do the numbers from the trinomial (the product) have to do with the numbers from the two binomials (the factors)? Two things, actually! 3 + 5 gives us __________, which was the coefficient from the ____________________ term from the trinomial. And 3 * 5 gives us __________, the ___________________ on the end of the trinomial. This means that if you can find two numbers that add to the _____________________ term of your trinomial and multiply to the _______________________ on the end, those are going to be the two numbers in your factored expression. Let's see if we can apply this idea to a different problem; maybe factor x2 + 11x + 24. So, for our method to work, we need to find two numbers that add up to the _____________________ coefficient (________) and that will also multiply to the _____________________ on the end (________). I like to write out all the different ways we can multiply to get the constant on the end and then see which one of those will add up to the number in the middle. In this case, (1 * 24)= 24. 24 is even, so _________ goes into it as well, so (_________ x _________), I think 3, (_________ x _________) and 4, (__________ x _________), and that's it. So we have these four options - which ones add up to 11? Hey, _________ and __________! That means that x2 + 11x + 24 factors into (x + ________)(x + _________). If you'd like to check your answer, you can quickly multiply it out and make sure it ends up back where we started!

3, 5, 8, middle, 15, constant, middle, constant, middle, 11, constant, 24, 2, 2, 12, 3, 8, 4, 6, 3, 8, 3, 8,

Once we get really good, we can even come up with the rule for the nth term simply given any two random terms of a sequence. Let's say that the fifth term is 22 and the twelfth term is 64. Apparently in 7 steps, going from the fifth term to the twelfth, I go up _________ points. That means I can do a quick _________________ problem to tell me that if it takes me 7 steps to go up 42, I must be going up ________ each step, which would make my common difference _________.

42, division, 6, 6

We said that there were 35 seats in the first row and four more in each row behind. That made the number of seats in the nth row an = _____n + ____________ because the zeroth row, if there was one, would only have 31 seats, and the common difference was __________. What we're really asking is what is the sum of a series that starts at the first row and goes through the 96th row while following the sequence 4n + 31? Again, the number of seats in this sequence is represented using summation notation just like this. So, how do we use our formula? Well, first we have to decide how many groups there are going to be. There are a total of 96 rows, so we have to divide that by 2 since we're taking pairs, so that's the _______/________ part of the formula. Now we have to find out how much each group is equal to. I find that by taking the number of seats in the ____________ row and the number of seats in the ___________th row before adding that together. For the first row, I can plug in _________ into that 4n + 31 formula, which gives me ___________. For the 96th row, I plug in _________, which gives me ____________. That gives me _______________ for each group. 96/2 gives me _____________ groups. So, 48*(450) = ________________, and we now know that there are _______________ seats in each section of Michigan Stadium!

4n, 31, 4, n/2, first, 96th, 1, 35, 96, 415, 450, 48, 21,600, 21,600

2x (3x - 5) = x + 30 This one isn't even equal to zero, but that's okay! We still need to use algebra to change this quadratic back to standard form. Our standard form quadratic equation will be __________^________ - ____________- _____________ = 0. Let's go ahead and use the quadratic formula for this one. That means we have to first correctly identify a, b, and c as the coefficients on our trinomial. A, the first coefficient, is __________, the next one, b, is __________, and the constant on the end, c, is __________. We now need to substitute these values into the formula, and now we need to follow the order of operations very carefully to come up with our two answers. Working through this one step at a time, the - -11 will turn into a __________, the inside of the square root (what's called the discriminant) has an exponent that we can do first. ___________ * ___________ is ___________. Then 4 * ___________ * ___________ is _____________. Continuing the order of operations, doing the multiplication on the bottom (2 * ________) would give me ___________. Doing a minus negative turns into plus a positive, which means we end up with 11 +/- the square root of 841/12, and it turns out the square root of 841 is ____________, which means I can split the two answers up into 11 + (29/12) or 11 - (29/12). Doing that and then simplifying our fractions gives us our two answers, ___________/___________ or ____________/____________.

6x, 2, 11x, 30, 6, -11, -30, 11, -11, -11, 121, 6, -30, -720, 6, 12, 29, 10, 3, -3, 2

Convert 834,000 to scientific notation. Even though there is no decimal point showing in this number, we know that it is at the end of the number. To convert the number to scientific notation, the first step is to move that decimal point from the end of the number to after the first non-zero number - in this case, the ________. Then we drop the trailing zeros, and the first part of our scientific notation is ______._________. To find the second part of the scientific notation number, count the ________________ of spaces that you moved the decimal point. For this example, the decimal point was moved ________ spaces. That number will be the exponent. And in this case, it will be ____________________ because the number is a large number. So the second part of the scientific notation for this example is x 10^_______. Putting it all together gives us 834,000 = _______.__________ x __________^_________.

8, 8.34, number, 5, positive, 5, 8.34, 10^5

Quotient of Powers Remember, 'quotient' means 'division'.' The formula says (x^______) / (x^________) = x^(_______ - _______). Basically, when you divide exponentials with the ______________ base, you ______________________ the exponent (or powers). Let me show you how this one works. Let's say I had (x^4) / (x^3). In the top (or numerator), we have x times x times x times x. In the bottom (or denominator), we have x times x times x. Hopefully, you remember that x divided by x is 1, so the xs cancel. So, x divided by x is 1, x divided by x is 1, and x divided by x is 1. So, when we cancel them, what are we left with? That's right: x^1, or just x. So (x^4) / (x^3) is just x^(_______ - _______), which is x^________.

a, b, a, b, same, subtract, 4, 3, 1

Negative exponents are a bit different. They still indicate how many times to multiply a number to itself, but it also means that the base is on the wrong side of the fraction line, which means that if the negative exponent is in the numerator (top of a fraction), to make it positive, it needs to be switched to the _________________________________ (bottom of a fraction). 3^-2 = __________ / ________^________ The reverse is also true. If the negative exponent is in the denominator, to make it positive, just move it to the __________________________. 1 / 2^-5 = _________^__________ 3^-3 can also be written as _________ / ________^________ or ________/_________. It can also be written as a decimal. To write a fraction as a decimal, you just need to divide - in this case, divide _________ by __________ to get _________________.

denominator, 1, 3, 2, numerator, 2, 5, 1, 3, 3, 1, 27, 1, 27, 0.037

If the terms you are working with have ___________________ ______________, there is not much you can do to simplify the expression. The only exception to this rule is if both the bases are _________________________. Then, to simplify, you can simplify each _____________ and ___________________ them together. For example, Simplify (2^3)*(6^2) 2^3 = ________ and 6^2 = _________ You can simplify this problem by multiplying 8 and 36. 8 * 36 = ______________

different, bases, numbers, term, multiply, 8, 36, 288

A factor is a term that can be extracted from the ___________________________. Think about the number 6. Its factors are __________ and __________. Why? Because __________ * ___________ is 6. With 4x - 8, we can extract a ___________. Each term is a multiple of __________. If we factor out a 4, we have _________(__________ - __________). What if we had 3x - 8? Is there a common factor? No. 3 and 8 are what we call relatively prime. Remember that a prime number has no factors other than ________ and _______________. Relatively prime numbers have no shared factors other than ________.

equation, 2, 3, 2, 3, 4, 4, 4, x, 2, 1, itself, 1

Let's say that we're asked to express the series 5 + 10 + 20 +40 + 80 + 160 + 320 + ... using summation notation. The first thing we should try to figure out is the rule for this series. I know that this is a __________________________ series because each term is multiplied by _________ to find the next one. 5 * ________ = 10, 10 * ________ = 20, and so forth. I know that any geometric sequence has the rule an = ________ * r^(____-_____). a1 in this case is _________, because this is where my sequence begins. r is _________, because I'm multiplying by ________ each time. So, the rule for the nth term is _______(_______)^(n - 1). And we can take this rule and slide it in directly to the right of this sigma. Now, I simply need to label the beginning and ending value of the series. We start with the first term, so I put n = _____ on the bottom. Because there is a '...' at the end of our series here, that means it goes on forever. So, there's actually no ending point, so we write an _____________________ above the sigma to indicate that this is an infinite series.

geometric, 2, 2, 2, a1, n-1, 5, 2, 2, 5, 2, 1, infinity

Let's start with numbers. Here's an expression: 6x + 12. It's a nice expression. To factor a number out of an expression, we need to find the _______________________ ______________________ factor. That's the __________________ factor shared by all the terms. The highest common factor here is __________. __________ is our highest common factor. What happens if we factor out a 6 from both terms? This means we divide each term by 6. 6x becomes just _________. 12 becomes _________. We write our factored expression as _________ (_________ + _________). Here's an expression: xy + 7y. Note that we can't factor out any __________________. But what is shared by both terms? They both have a _________. So, we can just pull out that _________. What happens if we do? The xy becomes just _________. And the 7y becomes just _________. So we have ________(________ + _________).

highest common, largest, 6, 6, divide, x, 2, 6, x, 2, numbers, y, y, x, 7, y, x, 7

An exponent is a number indicating how many times a number is multiplied by ____________________. With negative numbers, always check whether the exponent is odd or even. With odd exponents, the number will stay _____________________. With even exponents, the number will be _____________________. Solve x^4 when x= 6, x= __________ Simplifying Fractions Solve 5^8/ 5^5: simplifying first will save us some effort. Let's think about what we have here. The numerator is 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5. The denominator is 5 * 5 * 5 * 5 * 5. That's a lot of 5s. But you know what? We can cross out the duplicates. 5/5 is just ________. Let's cross out all five of the 5s on the bottom and five matching 5s on the top. That leaves us with one in the denominator and ______ * ______ * _______, or _______^_______, in the numerator. Well, at least we got the one in the bottom bunk to take its hat off. What's 5 * 5 * 5? Well, 5 * 5 is ____________. And 25 * 5 is ___________. So, 5^8/5^5 is ___________.

itself, negative, positive, 1,296, 1, one, 5, 5, 5, 5^3, 25, 125, 125

A sequence is just a pattern, and a geometric sequence is a pattern that is generated through repeated ________________________________. Each new term is made by _________________________ the previous one by the same thing over and over. The general rile for the nth term of a geometric sequence is an= a1 * r^n-1 where an=___________________ a1=____________________ r^n-1=________________ an is the ______________________ nth term, a1 is the ___________________term, r is the ______________________ _________________, or the amount that we multiply by every step of the way. It's called the common ratio because if you divide any two consecutive terms you'll get the same thing. Lastly, the n in the n-1 power is whatever _____________________ you're trying to find out.

multiplication, multiplying, general term, first term, common ratio, general, first, common ratio, term

I think the zero exponent is really fun! Basically, the formula says that anything - any number, any letter - raised to the zero exponent is always ______________. The Quotient of Powers Rule says that when we divide exponentials with the same base, we ________________________ their exponents. So, I have (x^3)/(x^3). Remember, we subtract their exponents. So that would be x^(3-3), which is x^___________. But where does the 1 come from? We have (x^3)/(x^3). That is, (x*x*x) on the top, or numerator, and (x*x*x) on the bottom, or denominator. x/x is ________, x/x is ________, and x/x is _________. When we multiply those together (1*1*1), we get ________! So that pretty much proves it: x^0 = 1!

one, subtract, 0, 1, 1, 1, 1

When you have two exponential expressions that have the ______________ ____________, you can easily multiply them together. All you have to do is ____________ the exponents. Here's an example: (3^2)*(3^5) To simplify this expression, just add the exponents: ______ + ______ = _______, and your answer is: ______^______ The rule also applies if one or more of the exponents are negative. Simplify (4^-3)*(4^5) Again, just add the exponents. ________ + ________ = _______ So, (4^-3)*(4^5) = _______^_______

same, base, add, 2 ,5, 7, 3, 7, -3, 5, 2, 4, 2

Power to a Power We can see from the formula we have (_____^______)^______. When you have a power to a power, you ___________________ the exponents (or powers). Let me show you how this one works. If I have (x^2)^4, which would be x^2 multiplied four times, or x^2 times x^2 times x^2 times x^2. Once again, we add all the exponents and get x^_______, and x^8 is the same as x^(_________ * _________), which is 8.

x, a, b, multiply, 8, 2, 4,

Power of a Product The formula says (x______)^_______ = (x^______) and (y^_______). When you have a product of a power, you give each base its own ________________. Think about it as distribution putting the exponent with each base. Let me show you how this one works. Let's say we had (xy)^2. That means we take _______ and multiply it twice that means _______ times _______. Well, that would give us two xs, or _______^_______, and two ys, or _______^________. That is the same as if I distributed _______ to the x, getting x^2, and ________ to the y, getting y^2.

y, a, a, a, exponent, xy, xy, xy, x, 2, y, 2, 2, 2