Math

14²

196

2/5

.4

3/5

.6

5/8

.625

2/3

.666 repeating

4/5

.8

5/6

.8333 repeating or .8334

7/8

.875

E + E =

Even 16 + 34 = 50

E x E =

Even 2 x 12 = 24

E - E

Even 32 - 8 = 24

Integer Quotient

(Q) The integer quotient is written in mixed numeral form (or decimal). D/S = Q with a remainder of ______

a²-b²

(a+b)(a-b) (algebraic identity) ie: (x²-9) = (x+3)(x-3)

To multiply two algebraic expressions, each term of the first expression is multiplied by each term of the second expression, and the results are added, as the following example shows:

(x+2)(3x-7)= 3x² - 7x + 6x -14 3x² -1x -14

1/3

.333 repeating

3/8

.375

Multiplying exponents

.03³ = .03 x .03 .03 So the product will be 6 places to the right of the decimal.

1/9

.1111 repeating (multiply by any number and it's that number repeating)

1/8

.125

1/7

.142857 repeating or .143

1/6

.1666 repeating or .1667

1/5

.2

¼

.25

4.) Rules of Exponents x⁰ =

1 ie: 7⁰ = 1 and -3⁰ = 1. 0⁰ is undefined.

O + O =

Even 7 + 9 = 16

Multiplying decimals - Part 1

1.) Count the number of digits to the right of the decimal point (ie: 6.25 x .048) Product will have 5 decimal places.

3/6

1/2 or .5

1.) Rules of Exponents x⁻ⁿ=

1/xⁿ ie: 4⁻³ = 1/4³ = 1/64 1/2⁻ⁿ = 2ⁿ

To solve a linear equation in ONE variable, simplify each side of the equation by combing like terms. Then use the rules for producing simpler equivalent equations.

11x -4 -8x = 2(x+4) - 2x 3x - 4 = 2x + 8 -2x 3x - 4 = 8 3x = 12 x=4

Another tricky question is, what is the smallest positive integer that when divided by 12 is a remainder of 5. For you may think that it's 17, but it's actually 5. Wait a second, what's going on here?

12 is bigger than 5, so if we divide 5 by 12, it goes into it zero times, an integer quotient of zero, and the remainder is 5. In general, if the divisor is larger than the dividend. Then the integer quotient is zero and the remainder equals the dividend. The test absolutely loves to test questions about this. 5/12 = 0 with a remainder of 5

11²

121

12²

144

13²

169

What's the only even prime number?

2!

Prime numbers less than 20

2, 3, 5, 7, 11, 13, 17, 19

Multiplying decimals - Part 2

2.) Then, ignore the decimal points and multiply. (In this case, we can use the DOUBLE and HALF method.) 625 x 48 1250 x 24 2500 x 12 5000 x 6 = 30,000

4/6

2/3 or .666 repeating or .6667

2/6

2/6 = 1/3 = .3333 repeating or .3334

15²

225

Multiplying decimals - Part 3

3.) Then, take the product and move the decimal 5 places to the left. 30,000 becomes .30000

O - O =

Even 9 - 5 = 4

A statement of equality between 2 algebraic expressions that is true for only certain values of the variables involved is called an equation. The values are called the solution to the equation. The following are some basic types of equations:

3x+5=-2 (a LINEAR equation with ONE variable) x-3y=10 (a LINEAR equation in two variables, x & y) 20y²+6y-17=10 A QUADRATIC equation in one variable, y.

A number of variable that is a factor of each term in an algebraic expression can be factored out, as the following example shows:

4x+12=4(x+3) 15y²-9y=3y(5y-3) 7x²+14x/2x+4 = 7x(x+2)/2(x+2)

Solving quadratic equations

A quadratic equation in the variable x is an equation that can be written in the form ax²+bx+c=0 where a, b, & c are real numbers and a≠0. When an equation has solutions, they can be found using the quadratic formula.

Rules of multiples 2.)

Another idea- if we need, say, the first. Five multiples of a number, we simply multiply the original number by the numbers 1, 2, 3, 4 and 5. So, suppose we needed the first five multiples with 12. So, that would be 12 times 1, 2, 3, 4 and 5. We can also get those by repeatedly adding 12. 12 plus 12 is 24. Plus 12 is 36. Plus 12 is 48 and so forth. So that's very interesting. We can add or subtract 12 and get new multiples of 12.

Is zero even or odd?

Even!

E x O =

Even. (With an E x O, there must be a prime factor of 2 included in the product, so must be even.) 6 x 9 = 56

a (b + c)

Distributive Property a × b + a × c

a (b-c)

Distributive Property a × b - a × c

D R --- = Q + --- S S

Dividend/divisor = Quotient + Remainder/ divisor

LCM = P x Q ------ GCF

Don't be tempted to multiply the top line first! Cancel before you do anything.

Prime factorization

Every positive integer greater than 1 must either be, A, a prime number, or B, be expressed as a unique product of prime numbers. So in other words, every integer greater than 1 that's not prime can be expressed as a, as a product of primes and this product is called the prime factorization of the number. So, here I've just shown an example of a very important idea. One way to say it is the following. The prime factorization of a number is like the DNA of the number, revealing all its essential ingredients. Any factor of Q must be composed only of prime factors found in Q. So that's a tricky idea.

To find the number of factors that a positive integer N has.

Find the prime factorization of N, and write it in terms of Powers of the prime factors, because we actually need those exponents. 8400 Step 2 is, create a list of the exponents. Remember to use 1 for a prime factor that has no exponent. 2⁴ x 3¹ x 5² x 7¹ 4 1 2 1 Once we have that list, step 3 is, add one to every number on the list. Creating a new list. 5 2 3 1 Once we have that new list, multiply them all together. Find the product of that new list. That product is the number of factors N has. 5x2x3x1 = 60 Therefore 8400 has 60 factors. To find the number of ODD factors, basically we would repeat this procedure, but ignore the factors of two. If we want the number of EVEN factors, we have to find all the factors, subtract the number of odd factors, and we discussed a little about why the procedure works.

Linear equations in TWO variables, x and y can be written in the form ax + by = c (Where a, b, and c are real numbers and a and b and not both zero.) For

For example, 3x + 2y = 8 is a linear equation in 2 variables. A solution of such an equation is an ordered pair of numbers (x,y) that makes the equation true when the value of x and y are substituted into the equation. For example, both (2,1) and (-2/3,5) are solutions of the equation 3x + 2y = 8, but (1,2) is not a solution. A linear equation in two variables has infinitely many solutions. If any other linear equation in the same variables is given, it is usually possible to find a unique solution to both equations. Two equations with the same variables are called a *system of equations*, and the equations in the system are called *simultaneous equations.* To solve a system of 2 equations means to find an ordered pair of numbers that satisfies *both* equations in the system.

GEMDAS

Grouping symbols; Exponents; Multiplication & Division; addition & subtraction. If there are multiple layers of parenthesis, we have to work our way from the outside in.

Rules of multiples 5.)

If P is a multiple of r, then any multiple of P is a multiple of r. ie: since 52 is a multiple of 13, you could add two 52s, 3 52s or any number of 52s 52 + 52 + 52 + 52 + 52 and it would still be a multiple of 13.

All even exponents mean this number a perfect square. k= 2⁶ x 3⁴ x 5²

If we see an unknown number in its prime factorization form, and all the exponents are even, we know right away the number must be a perfect square. For example, suppose we're given this number: k= 2⁶ x 3⁴ x 5² Well,we don't know what that number is, it's probably a big number. But we can tell right away just by looking at the prime factorization, that is definitely a perfect square because 6, 4, 2, all those exponents are even numbers.

Finding the Least Common Multiple (Least common denominator) Practice guessing and intuiting the Least Common Multiples to develop a math sense about these numbers!

If we wanted to find the LCM of a number, we could list: 8, 16, 24, 32, 48,→ 12, 24, 36, 48, 60 → In the case of 8 and 12; 24 is the least common multiple. Or, we could do the following: 1.) Find the two prime factorizations and the greatest common factor. In the case of 24 and 32: 24 = 2³ x 3 32 = 2⁵ Greatest common factor: 24: {1, 2, 3, 4, 6, 8, 12, 24} 32: {1, 2, 3, 4, 8, 16, 32} Greatest common factor = 8 Then take the GCF and use it to find what would need to be multiplied to get to the original number: 8x4 = 32 8x3 = 24 Then multiply the 3 different factors: 8 x 4 x 3 = 96. Therefore 96 is the Least Common Multiple or Least common denominator between 24 and 32.

exponent rule - true for all positive numbers x, except x=1 and for all integers a and b.

If x∧a=x∧b then a=b For example, if 2ⁿ=64 and 2⁶=64 therefore 2ⁿ=2⁶ and you conclude that n=6

Solving linear equations: The elimination method

In the elimination method, the object is to make the coefficients of one variable the same in both equations so that one variable can be eliminated either by adding the equations together or by subtracting one from the other. 4x + 3y =13 x + 2y = 2 In the example above, multiplying both sides of the SECOND equation by 4 yields 4x + 8y = 8 4(x + 2y = 2) 4x + 8y = 8 If you subtract the second equation from the first, *you elimate the x variable.* 4x + 3y =13 - 4x + 8y = 8 the result is -5y = 5. Thus y=-1 and subtracting -1 for y either of the original equations yields x=4. By either method the solution of the system is x = 4 and y = -1 or (x,y) = (4,-1).

Solving linear inequalities a.)

Inequalities can involve variables and are similar to equations, except that the two sides are related by one of the inequality signs instead of the equality sign used in equations. For example, 4x-1≤7 is a linear inequality in one variable which states that 4x-1 is less than or equal to 7.

Rules of multiples 1.)

Just as 1 is a factor of every positive integer, every positive integer's a multiple of 1. Similarly, just as ever positive integer is a factor of itself, every positive integer is a multiple of itself; those are two primary ideas.

Is 1 prime?

No!

2/x ≥ 1/3

Notice, first of all, that x cannot equal 0. If we made x equal 0, then we'd be dividing by 0, and we wouldn't have a sensible expression. Notice also that x has to be positive. Because, the fraction on the right is a positive number. And so the only way the fraction on the left is gonna be greater than it, or equal to it, is if it's also positive. There's no way that it could be negative. No negative can be greater than a positive. So x absolutely has to be positive. Because all numbers are positive, we can simply cross multiply. So, just plain old cross multiplying. We cross multiply, and we get that 6 is greater than or equal to x. Well, if we combine all these conditions together, we see that x has to be greater than 0 and less than or equal to 6. And so this would be the allowable range. Notice that there is an open circle at zero because that's not included, and there's a solid dot at six. 6≥0

E - O =

Odd 12 - 5 = 7

O + E

Odd 5 + 2 = 7

O x O =

Odd. (No prime factors of 2, so must be odd.) 5 x 5 = 25

Rules of multiples 4.)

Should mean I should be able to add any multiple of seven or subtract any multiple of seven and I will still have another multiple of seven. ie: 700 is a multiple of 7 49 is a multiple of 7 Therefore 749 should be a multiple of 7.

Counting factors in a perfect square

So, let's pretend that we're gonna count the factors in a perfect square. Notice, since all the powers of the prime factors of a perfect square are even, when we're counting the factors, the list of exponents will be a list of all even numbers. Then when we add 1 to each number on the list to get the second list, the second list will be a list of all odd numbers. Because of course when you add 1 to an even number you get an odd number. This means that the product of that second list will be a product of all odd numbers, so it will have to be odd. And, of course, the product of that second list is the number of factors, therefore, a perfect square always has an odd number of factors. That's a really big idea. Another way to see this is to think about factor pairs. Think about the factors of the perfect square 36. Think about the factor pairs. Well, of course one pair is 1 times 36. Of course, 1 if a factor of every number, every number is a factor of itself. 2 times 18, 3 times 12, 4 times 9, all these factors in pairs. But then we get down to 6 times 6, and of course, that counts as only one factor 6. Well, this is interesting. Every other factor was in pairs but 6, which is the square root of 36, is all by itself. And thus, we have an odd number of factors because we have one unpaired factor. And of course this is only gonna happen for perfect squares.

Solving quadratic equations by factoring

Some quadratic equations can be solved more quickly by factoring. For example: Here we need a number whose SUM is -1, and whose product is -6. 2x² - x - 6 = 0 (2x +3) (x-2) Then, set to 0 and solve for each. 2x-3=0 x-2=0 x=-3/2 x=2

Finding GCF (Greatest common factor/greatest common divisor) Practice guessing and intuiting the greatest common factors to develop a math sense about these numbers!

Suppose we want to find the Greatest Common Factor of 360 and 800. 1.) Find the prime factorizations: 360 = 2³ x 3² x 5 800 = 2⁵ x 5² 2.) What is the highest power of each number they have in common? 2 = 2³ 3 = 0 5 = 5¹ 3.) Find the product of these numbers Thus, 2³ x 5¹ = 8x5=40 40 is the greatest common factor/greatest common divisor.

Divisor

The number by which we are dividing (D/S)

Dividend

The number that is divided (D/S)

Solving linear inequalities c.)

The procedure used to solve a linear inequality is similar to that used to solve a linear equation, which is to simplify the inequality by isolating the variable on one side of the inequality, using the following two rules. • When the *same constant* is added to or subtracted to from both sides of an inequality, the direction of the inequality is preserved and the new inequality is equivalent to the original. • When both sides of the inequality are multiplied or divined by the same non-zero constant, the direction of the inequality is preserved if the constant is positive. **But the direction is reversed if the constant is negative.** In either case, the new inequality is equivalent to the original.

Solving linear equations: The substitution method

There are 2 basic methods for solving linear equations, by substitution or by elimination. In the substitution method, one equation is manipulated to express one variable in the terms of the other. The the expression is substituted in the other equation. For example, to solve the system of equations 4x + 3y =13 x + 2y = 2 you can express x in the second equation in terms of y as x = 2-2y. Then substitute 2-2y for x in the first equation to find the value of x. 4(2 - 2y) +3y = 13 8 -8y +3y = 13 8-5y=13 -5y=5 y = -1 Then, -1 can be substituted for y in either equation to find the value of x. We use the second equation: x+2y = 2 x +2(-1) = 2 x -2 = 2 x=4

Rules of multiples 6.)

To extend that, if P and Q are multiples of r then of course the product of P times Q must also be a multiple of r. That makes sense certainly because any multiple of P and any multiple of Q is a multiple of r.

Solving linear inequalities b.)

To solve a inequality means to find the set of all values of the variable that make the inequality true. This set of values of is also known as the *solution set* of an inequality. Two inequalities that have the same solution set are called *equivalent inequalities.*

How to test whether a larger number is prime.

To test whether any number is less than 100 is prime, all we have to do is check whether it is divisible by one of the prime numbers less than 10. And, of course, the only prime divisors less than 10 are 2, 3, 5, and 7. So those are the only divisors we have to check. It's not divisible by any of those numbers, it is a prime number.

Rules of multiples 3.)

We can actually generalize that idea, so that is multiple idea number three. If we know P is a multiple of r, then it must also be true that P minus r and P plus r are also multiples of r. If P and Q are multiples of r, then (P + Q) and (P- Q) must also be multiples of r.

(a+b)²

a² + 2ab + b² (algebraic identity)

Example of quadratic equation (p.226)

a²+b²+c=0 In the qaudratic equation 2x²-x-6, we have a=2, b=-1 and c=-6. Therefore the quadratic formula yields x= -(-1)±√(-1)² - 4(2)(-6) ------------------- 2(2) working with grouping symbols, we work under the square symbol first: x= 1±√49 -------- 4 x= 1±7 ------ 4 x= 1+7 ------ 4 x= 1-7 ------ 4 x=2, x=-6/4 or -3/2 Quadratic equations have at most two real solutions, as in this example. However, some quadratic equations only have one real solution.

(a-b)³

a³- 3²b + 3ab² - b³ (algebraic identity)

pos x neg

negative

.00001

one hundred thousandth = 1/100,000 = 10⁻⁵

.01

one hundredth = 1/100 = 10⁻²

.000001

one millionth = 1/1,000,000 = 10⁻⁶

.0001

one ten thousandth = 1/10,000 = 10⁻⁴

.1

one tenth = 1/10 = 10⁻¹

.001

one thousandth = 1/1000 = 10⁻³

neg x neg

positive

pos x pos

positive

Absolute value

the distance of the number from the origin

Quadratic formula

where the notation ± is shorthand for indicating two solutions-one that uses plus sign and the other that uses the minus sign.

2.) Rules of Exponents (x^a)(x^b)

x^(a+b) ie: 3² x 3⁴ = 3²⁺⁴ = 3⁶ y³ x y⁻¹= y³⁻¹= y² (Remember bases must be the same.)

3.) Rules of Exponents 3.) (x^a)/(x^b)

x^(a-b) = 1/x∧b-a ie: 5⁷/5⁴ = 5⁷⁻⁴ = 5³ = 125 and y³/y⁸ = y³⁻⁸ = y⁻⁵ = 1/y⁵ (Remember bases must be the same.)

7.) Rules of exponents (x^a)^b =

x^ab (4²)³ = 4⁶ (2⁵)² = 2¹⁰ = 1024 (3y⁶) = (3²) x (y⁶)² = 9y¹²

5.) Rules of Exponents (xⁿ)(yⁿ) =

xyⁿ ie: (2³)(3³) = 6³ = 216 10z³ = 10³ x z³ = 1000z³

x² - 16x + 48

x² - 16x + 48 sum = -16 product= 48 Therefore, both numbers will be a NEGATIVE. (x-4)(x-12). Notice that this method only works if the coefficient of x squared, the quadratic coefficient, is one. If the quadratic coefficient is something other than one, chances are very good that one of the other factoring methods can be used. Say, the difference of two squares or greatest common factor. It may mean that we have to factor out a greatest common factor.

6.) Rules of Exponents (x/y)ⁿ =

xⁿ/yⁿ (5/3)² = 5²/3² = 25/9 (r/4t)³ = r³/4t³ = r³/64t³