Powers and Roots

square root of positive integer and a fraction

If the base is greater than 1, then taking a square root makes things smaller, square root of b is less than b. If the base is between 0 and 1, taking the square root makes things bigger, radical b is greater than b. if b>1 , then if 0<b<1, then

rationalizing the denominator

If there's a single radical in the denominator, we rationalize simply my multiplying by that radical over itself.

square root or radical

Square roots and square root notation give us a way to talk about a situation in which the square of a variable does not equal a perfect square. Again, this is much more common than equaling a perfect square.

(radical 2)^48= 2^24

any even power of a square root can be written as a power of a whole number. So for example, if we had to deal with square root of 2 to the power of 48. Well all we have to do is really realize we can write that 48 as 2 times 24 and then bring that 2 inside the parenthesis. So this is root 2 squared, to the power of 24, which means that is 2 to the 24th. We can't compute the numerical size of that but we still can compare the size of that to something else.

a zero exponent

anything over itself must equal one. A to the 3 over a to the 3, that's something over itself that has to equal 1. So a to the zero, equals one

law of exponents I)

multiplying two powers: we just add the factors or the exponents when the base are the same it looks like you add them so Multiplying two powers of the same base, means that we can add the exponents.

summary

n summary, - when we add or subtract radical expressions, we simplify each term and combine terms with like expressions. - When we multiply or divide radical expressions we can treat the whole numbers and radicals separately. And we can multiply or divide two radicals right through the radical. - When we raise the radical expression to how we distribute the exponent to each factor and any even power of a radical is a power of a whole number.

Can we take the square root of a negative?

positive squared are positive, negative squared are positive, zero squared is zero. Nothing on the number line can be squared to yield a negative number

negative base less than -1

the absolute values of these powers are getting bigger each time. But the positive and negative signs are alternating. So this, this combines the idea of case one with continuously getting bigger. What's continuous getting bigger are the absolute values of the numbers. But the actual number itself, is flip flopping between positive, negative. So we get a big positive, then a bigger negative, then a bigger positive, then a bigger negative. It's going back and forth like that

Laws of exponents II

the distributive law says that P to the M plus or minus N, what we can do is just multiply the P separately times each one of those terms. That is the Distributive Law. Multiplication distributes over addition and subtraction. As it turns out, division also distributes over addition and subtraction. Much in the same way, exponents distribute over multiplication and division. So if I have a times b to the n, or a divided by b to the n, I can distribute the exponent to each factor. So a times b to the n equals a to the n times b to the n, a divided by b, that fraction to the n, equals a to the n divided by b to the n. So, we can distribute an exponent across multiplication or division. (ab)^n= (a^n)*(b^n) (a/b)^n= a^n/b^n you have to first find the prime factorization of the number that is a pair with the lowest prime factors

a power to a power. So I have a to the m, and that whole thing to the n.

the exponent has to be a to the m times n, and that's our law of exponents. so Raising a power to a power results in multiplying the exponents.

raising radical expressions to powers

the exponents distribute over multiplication. Thus if we're squaring a radical expression, we can square the number and the radical separately.

Equations with square roots.

- Certainly, we undo a square root by squaring, and we are always allowed to square both sides. Sometimes, for the simplest radical equations, all we have to do is square both sides. So, for example, if we have something like, square root of x plus 2 equals 3. Well, just square both sides, we get x plus 2 on the left, we get 9 on the right, subtract and we get x equals 7.

the patterns of exponential growth: positive base greater/less than 1

- So the big idea here is a positive base greater than one, the powers continually get larger, at a faster and faster rate. Numbers, when we have a base between zero and one, a positive base less than one. Then we are going to be following a very different pattern for exponential growth than if the base were more than one.

summary of Exponential Equations

- To solve exponential equations, we have to get equal bases on both sides. This may involve expressing the given bases as powers of smaller bases. - Once the bases on both sides are equal, we can equate the exponents and solve.

simplifying tips for the exponents of fraction

- first cancel out the coefficients of the numerator and the denominator so factor out the GCF of the two same bases

simplifying roots

- roots distribute over multiplication, so we can separate the root of the, of a product into the product of the roots. - First of all, remember that of course, it's easy to find the square root of perfect squares. ------------------------------------------- - we simplify square roots by factoring up the largest perfect square factor. - If we find the prime factorization or are given it, then we can use that. Any pairs of prime factors and any even powers of prime factors are perfect squares, and we can simplify those square roots.

summary of rationalizing

- to eliminate roots from the denominator of a fraction this is called rationalizing and it's something you always have to do. The test answers, the answers that appear on multiple choice will always be rationalized and so we will have to rationalize to match our answers to this. - If the fraction has a single root in the denominator, we rationalize simply by multiplying by that root over itself. - If the denominator of the fraction contains addition or subtraction involving a radical expressions, to rationalize we need to multiply by the conjugate of the denominator over itself.

ranking question tips

(1/3)^-8= reciprocal the inside and change the sign 3^8= Well what does that mean? That's the same as 3 to the positive 8. Now remember that 3 to the 4th is 81. So let's approximate that as 80. 3 to the 4th is approximately 80. Well 3 to the 8th is gonna be 3 to the 4th, squared. So that's gonna be approximately 80 squared. And 80 squared, that's up above 6,000 3^-3= 1/3^3= 1/27 (1/3)^5= 3^-5= 1/3^5 that is smaller than 1/3^3

3^32 - 3^28=

3 to the 28th is a factor of 3 to the 32nd. In fact, 3 to the 28th is the greatest common factor of these two terms. So we're gonna express both of them as products involving 3 to the 28th. 3^28 (3^4 - 1)= (3^28)(80)

the powers of two up to at least two to the ninth.

2^4=16 2^5= 32 2^6=64 2^7= 128 2^8=256 2^9= 512 and also 6^3=216 7^3= 343 8^3= 512 9^3= 729

square and cube roots

As with square roots, we can take any even root of a positive number, which results in a positive output. But we cannot take an even root of a negative number. We try to take the, an even root of a negative number, it is undefined. It does not equal anything on the number line. By contrast, as with cube roots, any odd root of a positive is positive and any odd root of a negative is negative. So it follows that same plus and minus pattern.

Important points

Exponents do not distribute over addition or subtraction. Those are very tempting mistake patterns and we can simplify the sum or difference of powers by factoring out the lower power. And finally, if we have bases are equal and we have a to the m equals a to the n we can equate the exponents.

It's important to be aware of a very common and tempting trap, because it's close to what is true.

First of all it's legal to distribute multiplication over addition and subtraction. That's 100% legal. P*(M+N)= PM + PN or P*(M-N)= PM - PN It's legal to distribute exponents over multiplication and division. That's 100% legal. (mn)^p= (m^p)(n^p) and (m/n)^p= (m^p)/(n^p) But it's illegal to distribute an exponent over addition and subtraction. (M+N)^p Not Equal M^p + N^p That is always illegal. And in fact m plus or minus n to the p means that we're taking that, what's in the parenthesis, m plus or minus n, and multiplying it by itself p times.

exponential equation

If two powers with the same base are equal, then the exponents must be equal. So if b to the x equals b to the y, it must be true that x equals y If the bases are not the same, you have to re-write the root of a side so as to make the bases identical. and the best tool is writing the root in a fractional exponent and then multiply by the variable outside. ------------------------------ Sometimes, neither of the basis can be written as a power of the other. Instead, both bases can be written as a power of some other, smaller number. This is actually the most common scenario in the test. Two bases and neither one can be written easily as a power of the other, but both can be written as powers of the third number. So we have to rewrite each base as a power of a common, smaller number. And then by using the laws of exponents we can get everything to equal bases and set the exponents equal. For example, if we had some power of 8 and some power of 16, we can't write 16 as a power of 8, we can't write 8 as a power of 16. We would have to begin by recognizing that both 8 and 16 can be rewritten as powers of 2.

summary of an equation with square roots

In summary, - to undo a radical equation, we need to square both sides. - We have to move something else to the other side sometimes, to isolate the radical before squaring. In other words, we need the radical by itself. So there are other terms on that side with the radical. We need to get rid of them, move them to the other side before we can square. - And the very act of squaring produces extraneous roots, therefore, we must check each answer the algebra gives us back in the original equation.

negative base between -1 and 0

It is between negative one and zero. we're getting closer to Zero but the +- signs are alternating

extraneous roots

It turns out that in radical equations we have to be aware of extraneous roots. When we do all of our algebra correctly, including squaring both sides of the equation, the algebra can lead to answers that don't actually work in the original equation. These are extraneous roots. IMP. when have complete our algebra, we then have to check our answers so as to make sure they work in the original equation. The only way we find is by plugging them in and check to see if the two sides of the equation are equal, if NOT, so the root does not work

When do you include the negative square root, when do you not include the negative square root?

It turns out that there's a very easy rule for determining this. If the radical sign is written by the test-maker, written as part and parcel of the way the question is asked, if that sign is printed as part of the question, then that means consider positive roots only. That's always what that sign means. But if the problem contains a variable squared, or your calculations lead to a variable squared, and you yourself have to initiate the act of taking a square root yourself. So in other words, the square root sign is not printed as part of the problem. You're going to have to take a square root yourself in the act of solving the problem, then you always have to consider both the positive and negative roots. That's the difference right there. Whether the radical sign is printed on the test or not

pattern mistakes

It's good to know, not only the patterns of what is true, but also the typical mistake patterns because the test always likes to test those particular mistake patterns. First of all, notice that all these laws work if the bases of the two powers involved are the same. We cannot apply any of these rules if the bases are different. - Finally, there is no law for the sum or difference of powers. So if we're adding three to the fourth plus three to the seventh or 5 to the eighth minus 5 squared. There is no fixed pattern for this. There is no single law of exponents. In a couple lessons, we'll learn how to use factoring out to simplify this kind of situation. So in other words, we may see this, but it's not a simple law of exponents.

What is the sum of the digits of integer x, where x = 4^10 x 5^13? (A) 13 (B) 11 (C) 10 (D) 8 (E) 5

Notice that 4^10 can be rewritten as 2^20. We can now express x as 2^20 x 5^13. The logic here is that 2 x 5 = 10. That is, 10 to any integer power greater than 1,will be a 1 followed by zeroes So now let's rewrite the problem again so we get 2^7 x 2^13 x 5^13. Combine 2^13 x 5^13 and we get 10^13. That is, we get 1 followed by 13 zeroes. If you are taking the sum, it's straightforward: 1 plus 13 zeroes is 1. We are not done yet as we have the 2^7. When you multiply this out, you get 128. 128 x 1,000 thirteen zeroes is equal to 128 followed by the thirteen zeroes. Ignore the zeroes and we get 1 + 2 + 8, which equals 11. Answer B

(X-1)^2= -4 (X-4)^3= -1

Notice, also, that an equation of the form something squared equals a negative has no solution. But, we could have something cubed equals a negative, that's perfectly fine. If something cubed equals negative one, then that thing must equal negative one and then we can solve for x. X-4= -1 so X= 3

one big mistake in simplifying the radicals

One big mistake is to multiply a whole number through a radical sign, to the number on the inside of the radical.

What happens if the exponent is not an integer, but a fraction?

Raising something to the power of one-half is the same as finding the positive square root of it. - And I will say if you actually have to do a calculation, if you have to actually choose between these two, always make things smaller before you make things bigger. - the denominator is behind the radical and the numerator inside the article as an exponent of the value inside

Operations with Roots / Multiplying and Dividing radical expressions

Remember first of all that multiplication is commutative and associative. What does this mean? It means when we're multiplying things we can swap the order around in any way we like. - we're simply gonna group the whole numbers together, multiply whole numbers by whole numbers. And group the radicals together, multiply radicals by radicals Sometimes we get a product in the radical that we can simplify IMP. the product of radicals can almost always be simplified when the radical expressions are big numbers just find factors of both -------------------------------------- we divide we divide whole numbers by whole numbers and we divide radicals by radicals right through the radical signs.

Operations with Roots / adding and subtracting radical expressions

The test will expect us to do all kind of arithmetic with roots and radical expressions. doing your arithmetic with radical expressions - we can not add or subtract directly through the radical signs. And it's very important to appreciate this, because, when your mind is under pressure in the test, your mind is gonna be drawn to making a mistake like this. So it's especially important to be clear about this, so that you don't make this mistake when you're under pressure. Instead when we have a sum or difference of radicals, we have to simplify each radical separately, by itself. And then we can add or subtract the ones that have the same radical factor. - NOTICE that in order to simplify the number under the radical, write in the form of prime factorization

cube roots

We can take the cube root of any number on the number line, positive, zero, or negative. So unlike the square root, the cubed root can have a negative output. When we put in a negative we get out a negative

Very IMP point in solving the equation with square roots

We need to square both sides to undo the radical, but this very act can produce extraneous roots. If we get a quadratic after squaring, which is common on the test, the algebra will lead to two roots. Sometimes both roots work. Sometimes one root works, and one is extraneous. Sometimes both are extraneous, and the equation has no solution ----- we completed an equation and X=-2 when we checked it or plugged -2 in the original equation results in the square root of a negative on both sides. So, we get the square root of negative 6, and square root of negative 6 is something outside the real number system, it does not live anywhere on the number line. So we can't do math with that. That is just, for our purposes, that is just an error and this equation has no solution.

For example, consider x squared equals two. X^2= 2

What number squared would equal 2? This would be a decimal between 1 and 2. We denote the positive number that, when squared, would equal 2 as, we use this notation. We read this either as the square root of 2 or radical 2.

when happens when we divid the powers? second law of exponent

When we divide powers of the same base, this means that what we have to do is subtract the exponents.

Notice

When we read this equation from left to right, we say that we are distributing P. When we read this equation from right to left, we say that we are factoring out P P * (M +- N) = PM +- PN It's also important to remember that any higher power of a base is divisible by any lower power of that same base. Thus, in the sum of a higher power and lower power of the same base. The greatest common factor of the two terms is the lower power, power, and this can be factored out because the lower power is always a factor of a higher power.

square roots of + and -

With square roots, we can find the square root only of positives. Of course, we could also find the square root of zero, but we cannot take the square root of a negative

Notice that as the exponent increases, whether the power gets bigger or smaller depends on the base.

X^7 > X^6? we asked the question is x to the seventh greater than x to the sixth? Well there is no clear answer. It would be true for positive numbers greater than one and false for negatives. Also if x equals zero, x to the 7th would equal x to the 6th which would be zero. let's suppose X<1, and X is not equal to zero, so the answer is X^7 < X^6 First of all it's very easy to think about what happens with the negatives. If x is negative then x to the seventh is negative and x to the sixth is positive. And any positive is greater than any negative. So therefore we're gonna get a no answer to the question. We're gonna get a clear answer of no. X to the sixth is definitely gonna be bigger if x is negative. Of course, x to the sixth is going to be bigger than x to the seventh.

negative exponents

b^-n= 1/b^n And this suggests that b to the negative n equals 1 over b to the n. So, that is the exponent rule. That is the rule for negative exponents. yet another way to think the negative exponent So, the rule, of course, is b to the, to the negative n equals one over b to the n. So, another way to say this is a base to a negative power is the reciprocal of that same base to the positive power. This means that a negative exponent on a fraction will be the reciprocal to the positive power. (p/q)^-n = (q/p)^n Notice A negative power in the numerator of a fraction can be moved to the denominator as a positive power, or likewise, from the denominator to the numerator.

summery of Negative exponents

b^-n= 1/b^n A base to a negative exponent is one over the base of, to the positive of that exponent. A fraction to the - n equals the reciprocal, to the + n. So we could flip over a fraction and get rid of the negative in the exponent. And exponents switch from negative to positive when we move them in a fraction from numerator to denominator, or vice versa. . and their coefficients will be cancelled out if there is a GCF

exponents basic rules

b^n, means n factors of b multiplied together One to any power is one. Zero to any positive power is zero. A negative to, to an even power is positive. A negative to an odd power is odd. An equation with an expression to an even power equal to a negative has no solution, but an odd power can equal a negative.

tips of simplification about law II

first you have to distribute the terms with the whole powers so as to throw the parentheses away. power to power means multiply the exponents y^12/y^-5= y^12 - (-5)= y^17

compare the size of different order roots

for a number greater than one, the higher the root, the smaller the actual number. Suppose we're taking different roots of 19. Well of course the square root of 19 has to be smaller than 19. Now the cube root of 19 has to be even smaller That is, with numbers bigger than one, taking higher order roots, make it smaller and smaller, and move it closer to one. Taking root of numbers between zero and one makes it bigger, and they move up closer and closer to one. So this is the pattern. Everything gets closer to one. And one of the reasons for that, of course, is that we can take any root of one and it equals one.

tip of equation with square roots

keep in mind that we should square both sides only when the radical is by itself on one side of the equation. If the radical appears on, with other terms on one side, we will have to isolate the radical on one side, before it would make sense to square both sides

how to rationalize a fraction with a radical expression involving addition or subtraction in the denominator.

multiply the denominator by its conjugate, and the rest of the story

summary of square root

unlike with square roots, we can take cube roots of both positive and negatives. That's a big idea. In fact, we can take any even root of positives only, not negatives, but we can take any odd root of any number on the number line. That's also a really big idea. Any root of one equals one, any root of zero equals zero. All roots preserve the order of inequalities, assuming all the numbers are positive. And the higher the order of a root, the closer the result is to one. So again, the numbers larger than one, when we take roots, they get smaller, move closer to one. When we take roots of numbers that are between zero and one, they get bigger, and they move closer to one.

17^30 + 17^20 =

we know that 17 to the 30th has to be divisible by 17 to the 20th. We know one is a factor of the other. And so 17 to the 20th is the greatest common factor of these two terms. So I'm gonna factor that out. 17^20 (17^10 + 1)