Volume 2 - Number Properties & Algebra

Dividing terms with the same exponent

divide their bases ex: 10^4 / 2^4 = 5^4

Zero to any power...

equals 0 (except 0^0, which is indeterminate)

Prime Factor

factors that are prime numbers

Multiples

integers that can be divided evenly by "n" - they can be negative - zero is a multiple of every integer - every number has MORE multiples than factors

Exponent Rule: Multiplication: Same Base, Same Exponent n^5 x n^5

n^10 or (n^2)^5

Exponent Rule: Division: Same Exponent a^7 / b^7

(a/b)^7

Exponent Rule: Multiplication: Same Exponent f^4 x g^4

(fg)^4

Exponent Rule: Difference of Squares x^2 - y^2 101^2 - 99^2

(x+y)(x-y) (101 + 99)(101-99) = 200(2) = 400

Exponent Rule: Multiplication: Same Base n^3 x n^3 3^5 x 3^5

n^3 + 3 = n^6 3^5 + 5 = 3^10

RARE Root Rule: Decimals in Roots Cube root(.000064)

.04, since .04 x .04 x .04 = .000064

1^3

1

Square root of 1

1

RARE: Advanced Shortcut: x-1 vs. 1-x

1 - x is simply x - 1 multiplied by -1 1 - x can be factored as -(x-1) (same with 3 - n, 2 - 5y, etc.)

Flipping the base: x^(-n)

1 / x^n This is how you can make any exponent positive by taking the reciprocal of its base **you can also flip its base to the top if you have something like this: 1/(3^-2) --> 3^2 --> 9 ** you can also flip bases top and bottom if you have something like (3^-1)/(4^-1) --> 4/3 **be sure you only flip the base and not the coefficient 2(x^-3) --> 2/x^3

RARE: To determine number of non-zero digits within a decimal

1. Convert the "10"s in the denominator to 10^-n 2. Express the remaining fraction as a decimal 3. The non-zero digits of that decimal will be the non-zero digits you are looking for Ex: x = 1 / 2^11 x 5^8 x contains 7 "10s" in its denominator and then 1 / 2^3 x 10^8 = 1 / 2^3 x 10^-8 --> 1/8 x 10^-8 = 0.125 x 10^-8 So there's a total of three non-zero digits b/c .125 Another example is 1 / 2^10 x 5^ 15 which would give you 1 / 5^6, which is a huge number so instead you can do 0.2^6 which is .000064, so that's two non-zero digits

Counting factors (determining total number of factors for an integer)

1. Determine the integer's prime factorization 2. Add 1 to each exponent within that prime factorization 3. Multiply the resulting sums Ex: 550 Prime factorization: 2^1 x 5^2 x 11^1 2, 3, 2 2 x 3 x 2 = 12 550 has a total of 12 factors **This is rare

To solve a quadratic equation...

1. Factor it 2. Complete the Square 3. Use the quadratic formula *For the GRE, you only have to know how to factor

Perfect Cube

number whose cube root is a whole number

Perfect Square

number whose square root is a whole number

Factors (divisors)

numbers that divide evenly into a given number - they cannot be negative

Cube roots can only be...

positive

To find, for example, integers that leave a remainder of 4 when divided by 5...

1. Generate three multiples of your divisor 2. Add the remainder to each 5+4 = 9 10 +4 = 14 15 + 4 = 19

Symbolism Questions (function questions containing weird symbols)

1. Identify the definition 2. Treat it as a formula Sometimes the question will also involve outputs - you just plug in the numbers at different points.

To solve for variable x

1. Identify what you want to remove to get x alone 2. Remove it by doing the opposite (using PEMDAS backwards) 3. Do the opposite operation to both sides

RARE: Combining Inequalities and Equations

1. Isolate a variable they share in common 2. Substitute the equation into the inequality

Absolute Value Equations/Inequalities

1. Isolate the absolute value brackets (if necessary) 2. Drop the brackets and place +- in front of the algebra -- +-(Algebra) =/</> Answer 3. Then solve the two equations that result BEWARE: Both solutions are not always valid. Plug both of them back into the question to ensure that each works.

To determine GCF and LCM of big numbers

1. Make prime factorization tables and organization in rows 2. To determine GCF, identify factors in common in each column and multiply 3. To determine LCM, identify the largest factors in each column, and multiply **This is rare on the GRE

Remainders are...

1. Never negative 2. Always smaller than their divisors

Solving "plug in" questions

1. Plug in the numbers (generally easier) 2. Manipulate the algebra (difficult but can be faster - if you like math) Before you plug, remove any terms shared by the two expressions. Goal is to prove an inconsistent relationship between the two quantities Test at least three numbers (different numbers) In general, test 1, 0, and -1 always. If the problem restricts this, consider extreme values such as the values in between -1 and 1, or very large or small numbers. If one set of numbers makes A larger and another set makes B larger, the answer will always be D - cannot be determined

The Denominator Trick

1. Put parentheses around the entire equation 2. Multiply it by the denominator you wish to eliminate *Use this if either side of the equation is not a fraction

Factoring a quadratic equation...

1. Set the equation equal to 0 2. Make sure the x^2 term is positive 3. Set up two set of parentheses like (x )(x ) = 0 4. Complete the parentheses by choosing two numbers, which should multiply to the last value in your quadratic but add to the middle value 5. Set each factor equal to 0 to solve for roots

RARE: Combining Inequalities

1. Stack the inequalities on top of each other 2. Make sure the inequality signs point in the same direction 3. Add them You don't need to eliminate variables like in equations.

The only two ways to generate odd number through +, -, and x

1. The addition/subtraction of one even number and one odd number 2. The multiplication of two odd numbers

Two facts about prime numbers

1. The number of prime numbers is infinite 2. The square of any prime number has exactly 3 factors

When factoring...

1. What is the largest element in common in all of the terms? 2. What is multiplied to create the original expression? It is usually in your best interest to factor any expressions that can be factored and to distribute expressions that can be distributed.

Like terms

same base and same exponent x^2 + x^2 x^2 = 3x^2 6y^3 - 2y^3 = 4y^3

Square root of 2

1.4 (approx)

Square root of 3

1.7 (approx)

Exponent Rule: Negative Exponents: Flip The Base x^-2 , 1 / x^-2 5^-2

1/x^2, x^2 1/5^2

Root Rule: Division 10Root(15) / 2Root(5)

10/2 x Root(15/5) = 5Root(3)

10^2

100

10^3

1000

11^2

121

5^3

125

Root Rule: Multiplication 3Root(p) x 4Root(q)

12Root(pq)

Perfect square

square of an integer

To compare two roots...

square them (the whole thing - even coefficients, etc.) *Particularly effective for comparing quantities involving roots within roots - keep squaring both sides until all roots are gone *This technique only works for positive values, so be careful of squaring variables because they can be negative or positive

12^2

144

2^4

16

4^2

16

13^2

169

14^2

196

2^1

2

Square root of 4

2

The only even prime number

2

Prime numbers under 30

2, 3, 5, 7, 11, 13, 17, 19, 23, 29

Square root of 5

2.2 (approx)

Square root of 6

2.4 (approx)

Square root of 7

2.6 (approx)

Square root of 8

2.8 (approx)

Base

term to which an exponent is attached Ex: 2y^3 y = base

6^3

216

15^2

225

3^5

243

4^4

256

3^3

27

Like terms

terms that share the same variable and the same exponent

Dividend

the integer that gets divided

Greatest Common Factor (GCF)

the largest factor shared by two or more integers

The more we multiply numbers between zero and one...

the smaller they get

Least Common Multiple (LCM)

the smallest integer divisible by two or more integers

Imaginary number

the square root of a negative number

RARE: Integer Constraints

2a + 3b = 5 --> Normally, it is impossible to determine a and b. However, if they must be integers, it MAY be possible to solve for them. You need to know the ADDITION OR SUBTRACTION MULTIPLE RULE: If two terms in an equation are multiples of M, then the remaining term in that equation must also be a multiple of M. Ex: 49 - x = 14. 49 and 14 = both multiples of 7, so x must be too. Ex: If a and b are positive integers, and 5a + 7b = 63, then a + b = ? 7b and 63 are both multiples of 7. 5a must also be a multiple of 7 5(7) + 7b = 63

3^1

3

Square root of 9

3

Square root of 10

3.2 (approx)

2^5

32

Equation

3x -6 = 9 Only equations can be used to solve for x

2^2

4

4^1

4

20^2

400

Root Rule: Addition/Subtraction 6Root(7) - 2Root(7)

4Root(7)

RARE Root Rule: Roots as Exponents Root(4)

4^(1/2)

Expression

4x + 7

25^2

625

2^6

64

4^3

64

2^3

8

3^4

81

3^2

9

30^2

900

Function (Function questions are rare on the GRE - maybe 1-2 per test)

A formula for which any input (or set of inputs) has exactly one output Ex: 2x = y

To multiply terms with the same base...

Add their exponents and make sure to NEVER multiply the base Ex: 3^4 x 3^2 = 3^6 *Coefficients of different bases can be multiplied Ex: 4x^2 x 7y^5 --> 28x^2y^5

Exponent Rule: Powers to Powers (w^4)^3 (2^3)^3

w^(4 x 3) = w^12 2^9

Integers

whole numbers (including zero)

Quadratic Equation

Any equation that contains a variable raised to both the first and second powers

Root Rule: Approximation Root(55)

Approx. 7.4, because Root(55) is slightly closer to Root(49) than Root(64)

Taking the root of a decimal (same goes for cubed roots)

Ask yourself WHAT x WHAT equals the value under the radical? OR Use the formula: "Decimal Places of Answer" x "Root Number" = Decimal Places of Radical Then ignore the decimal point within the radical and "take the root" of that value

Exponent Rule: powers of 0 x^0 (-7)^0

x^0 = 1 and (-x)^0 = 1, if x does not equal 0 (-7)^0 = 1 and -7^0 = -1

Exponent Rule: Powers of 1 x(x^4) 3^4 / 3

x^1(x^4) = x^5 3^4 / 3^1 = 3^3

Exponent Rule: Addition/Subtraction: Factor x^2 + x^3 2^6 - 2^5

x^2 (x^0 + x^1) = x^2(1+x) 2^5(2 - 1) = 2^5(1) = 2^5

Difference Between Squares Formula

x^2-y^2 = (x+y)(x-y)

Exponent Rule: Division: Same Base x^8 / x^3 5^8 / 5^3

x^5 5^5

If the order of the points A, B, and C on a line is not given...

you cannot assume a specific order

To resolve parentheses in a complex expression...

you have to combine the terms within the parentheses before distributing the exponent Ex: (x+y)^2 --> (x+y)(x+y) --> x^2 + 2xy + y^2

To add or subtract exponent expressions...

you have to factor them Ask: 1. What is the largest element in common to all of my terms? 2. What do I have to multiply with that common element to recreate the original expression? Ex: 12n^3 + 4n^2 + 8n --> 4n(3n^2 + n + 2)

If an equation contains even exponents...

Be careful because it can have two solutions: one positive and one negative. Odd exponents only have on solution: positive. Note: Taking the root of an expression with an even exponent technically results in its absolute value

Root Rule: Values from Root(2) to Root(10)

Benchmark: Root(4) = 2. Larger roots increase by 0.2, smaller roots decrease by 0.3

"Plug In" Questions

Comparison of algebraic expressions that can have limitless/nearly limitless values

Functions

Consist of a header, a formula, and an input f(x) = 4x^2 + 3, what is the value of f(-2)? f(x) = the header: the value of f is dependent on the value of x 4x^2 + 3 = the formula f(-2) = the input The formula + the input tell you what to plug in and where to plug it. Can also contain outputs sometimes.

Compound Inequalities

Contain more than one statement Ex: 8 > x > 4 *These are rare on the GRE

To solve a variable equation comparing two fractions...

Cross multiply (multiply denominator by opposite numerator) OR Change the denominators to a common multiple and then change the rest of the equation to match, the solve for the variable

When subtracting exponent expressions, watch out for the difference between two squares

Difference between square: x^2- y^2 --> (x+y)(x-y) 101^2 - 99^2 may look like it should be 2^2, but actually: (101 + 99) (101 - 99) = (200)(2) = 400

The Distribution Shortcut

Distribute the exponent to every term within the parentheses (3n)^4 --> (3^4) x (n^4) = 81(n^4) **only works for simple expressions (not containing addition or subtraction), not complex **ex of a complex: (x+y)^2

If your equation contains multiple denominators that you'd like to simplify...

Eliminate all of them by multiplying the equation by the lowest common denominator (LCD)

One to any power...

Equals one

RARE: Advanced Shortcut: Multiplying Equations

Equations can be combined with multiplication. The opportunity to do this isn't common, but ex: If abc can't = 0, and if r = ac/b and s = ab/c, then 1/rs = ? ac/b x ab/c = a^2bc/bc --> a^2

Is zero even or odd?

Even

Prime Factorization

Ex: 54 would be 2^1 x 3^3

Simplifying "Complex" Roots ex: Root(16(4) + 32(16))

Factor them - look for perfect squares

To determine the value of a larger root or cube root...

Find the perfect square that it lies between and then approximate

FOIL

First Inner Outer Last How to multiply something like (n+5)(n-2) If the factors of an equation are already set equal to 0, it is a mistake to FOIL it.

Irrational Fraction

Fractions with square roots in their denominators

RARE: Absolute Value Graphs

GRE may ask you to translate a number line into an equation or inequality involving absolute value. Use this formula, in which the midpoint represents the average of two endpoints: [] = absolute value brackets [x - midpoint] <= Distance from Endpoints to Midpoints

The number 0's three unique properties

If n is any number: 1. n +/- 0 = n 2. n x 0 = 0 3. Division by 0 is impossible. 0/n = undefined

RARE: Addition or Subtraction Multiple Rule [Integer Constraints]

If two terms in an equation are multiples of M, then the remaining term in that equation must also be a multiple of M.

To isolate a variable in a compound inequality...

Isolate the variable in the middle of the compound by doing the same thing to all three sides of the inequality

Multiplying terms with the same exponents

Just multiply their bases and leave the exponents alone - whatever's the same has to remain the same 2^4 x 5^4 = 10^4

When working with confusing/convoluted number line problems you should...

Label the intervals with variables

To express a root as an exponent...

Let a square root = an exponent to the 1/2 power (1/3 power for cubed, etc.)

To raise a power to a power... (Ex: (4^3)^3)

Multiply the exponents

To multiply roots with coefficients...

Multiply the roots and coefficients separately Ex: 5(root2) x 5(root2) = 5 x 5 x root2 x root2 = 25 x 2 = 50

How to simplify an irrational fraction

Multiply the top and bottom of the fraction by whatever makes the radical disappear In most cases, you can rid a denominator of its square root by multiplying the fraction by the same square root

Solving fractions with "complex" denominators (addition or subtraction)

Multiply them by their conjugates Then simplify by factoring the difference between the perfect squares

If raised to an odd exponent, negative one =

Negative one

Is 1 a prime number?

No

A variable with an even exponent typically has two solutions...

One positive and one negative square root of 25 is +5 or -5

A variable with an odd exponent...

Only has one solution (positive)

RARE: Functions & Sequences: Picking Numbers Problems with no concrete values Ex: For which of the following functions f is f(x-1) = f(1-x) for all x?

Pick numbers and keep them small and easy to work with (Plan B strategy) Monster formulas often lack concrete values and in many cases, one variable with the formula increases/decreases in value while another remains constant Ex: The quantities A and B are positive and are related by the equation A = x/b, where x is a constant. If the value of B increased by 25 percent, then the value of A decreases by what percent? A: Pick a value for the variable that changes, then pick another for the constant.

To divide two or more square roots...

Place the division within a single root (this also works the opposite way)

To multiply two or more square roots...

Place the multiplication within a single root (this also works the opposite way)

RARE: Advanced Shortcut: Avoiding Quadratics

Plugging an equation containing +/- into an equation containing multiplication will produce a quadratic equation if those equations have the SAME two variables. You can "avoid" the quadratic and do this faster: 1. Find two numbers that multiply to your product but add to your sum Ex: x + y = 6 and xy = -7 Ex: If v + w = 8 and vw = 12, which of the following are possible values of w?

If raised to an even exponent, negative one =

Positive one

Root Rule: Simplification Root(28)

Root(2 x 2 x 7) = 2Root(7)

RARE Root Rule: Factoring Complex Roots Root[9(4) + 9(5)]

Root[9(4+5)] = Root(9x9) = 9

Adding roots

Roots with the same radical can be added or subtracted by combining their coefficients. Roots with different radicals cannot be added or subtracted. Instead, simplify the roots if possible and then approximate.

RARE: Advanced Shortcut: The "Something" Shortcut

Say you have 6 - (x/5) = 4. You can say 6 minus SOMETHING equals 4. This holds true for more complicated problems, like: (8a)/(6 - 2b/3c) = 2a. 8a divided by SOMETHING equals 2a.

To graph an inequality on a number line...

Shade every point on the number line that is a valid solution to the inequality and leave open other points (filled circle = greater/less than or equal to) (unfilled circle = greater/less than)

The "FOIL" Identities **EXTREMELY important

Simple: (x+y)^2 Foiled: x^2 + 2xy + y^2 Simple: (x-y)^2 Foiled: x^2 - 2xy + y^2 Simple: (x+y)(x-y) Foiled: x^2 - y^2 (Difference between squares) Some hidden/scrambled forms examples: x^2 -1 --> (x+1)(x-1) x^4 - y^4 --> (x^2 + y^2)(x^2 - y^2) x^2 - 4x + 4 --> (x-2)^2 x^2 + y^2 = -2xy + 16 --> x^2 + 2xy + y^2 = 16 --> (x + y)^2 = 16

Root Rule: Simple vs. Complex Simple: Root(36 x 36) Complex: Root(36 + 36)

Simple: 36 Complex: Root(72)

Exponent Rule: Resolving Parentheses: Simple vs. Complex Simple: (xy)^2 ... (3x)^2 Complex: (x+y)^2 ... (4+9)^2

Solutions: Simple: (x^2)(y^2) ... (3^2)(x^2) = 9x^2 Complex: (x+y)(x+y) ... (13)^2 = 169

To divide terms with the same base...

Subtract their exponents and leave the bases alone *Coefficients of different bases can be divided Ex: 8a^4/12b^2 --> 2a^4/3b^2

Conjugate

Switching an expression's plus or minus sign

To simplify square roots... (also cube roots)

Take out anything that's a perfect square OR 1. Break the number into its prime factors 2. Replace any "pairs" with the same number (i.e., two fours in makes you write one four out)

When inequalities are multiplied/divided by a negative value...

The direction of their inequality sign must always be flipped *be careful of this with variables - when multiplying or dividing by a variable you may not know if it's negative or positive

Intervals vs. Tickmarks on a line problem

The number of tick marks will always be one greater than the number of intervals

Quotient

The resulting portion of the division that is not the remainder

RARE: Maximization problems

These questions ask you to identify the greatest/smallest value for a particular variable/expression. 1. Identify the greatest and smallest values that are possible for each input. 2. Test every combination of those inputs Ex: If -4 < x < 8 and -6 < 7 < 2, what is the greatest possible value for x-y?

Exponent Rule: Complicated Bases: Break Them Down 12^7

(2^2 x 3^1)^7 = 2^14 x 3^7

Sequences that are defined by the terms that precede or succeed them Ex: The sequence P1, P2, P3... Pn, is defined as Pn = Pn-1 + Pn-2 for all n>2. If P1 = P2 = 1, which of the following equals P5?

To solve: Translate the formula into a simple statement. Ex: Each term (Pn) after the second (n>2) equals the sum of the previous term (Pn-1) and the term before that (Pn-2). Another, trickier example: The sequence T1, T2, Tn... is such that Tn = Tn-1 + Tn-2 / 2 for all n >2. If T5 = 8 and T7 = 10 what is the value of T8? So each term after the second is equal to the average of the previous terms.

RARE: Functions & Sequences: Embedded Functions One function embedded inside another

To solve: 1. First solve the inside function 2. Insert the solution into the outside function to complete the problem

Sequence problems with no explicit formulas Ex: In the sequence of numbers A1, A2, A3, A4, A5, each number after the first is three times the preceding number. If A4 - A2 = 72, what is the value of A3?

To solve: 1. Label the lowest (or highest) term in the sequence x. A1 = x 2. Flesh out the sequence until you have labeled all the terms specified in the problem A2 = 3x A3 = 9x A4 = 27x A5 = 81x 3. Set up an equation to solve for x 27x - 3x = 72 --> 24x = 72 --> x = 3 So then 9(3) = 27

Function Question: Monster Formula Ex: Q = 2P/R+S or C = 2(pi)(x^2)/d

To solve: 1. Solve for a variable within the formula. Most times you can do this by plugging values supplied by the question in other variables. In more difficult examples, you may need to plug in twice in order to solve for a constant. Sometimes these problems may imply information rather than state it directly - especially geometry questions.

RARE: How many consecutive zeroes a number has to the left/right of its decimal point

Understand that zeroes come from "10"s or "2 x 5"s Ex: If k = 2^6 x 5^10 is expressed as an integer, how many zeroes will k have between its decimal point and its first non-zero digit? A) 6, because it contains six "2 x 5"s To determine the number of consecutive zeroes to the right of a decimal point: 1. Count the number of "10"s in its denominator 2. Determine the product of any numbers that do not produce a "10" 3. If it's less than 10, product will add zero "10s" 4. If it's between 10 and 100, product will add one "10" 5. If it's between 100 and 1,000, product will add two "10"s Ex: g = 1 / 2^6 x 5^7 x 7 --> 6 0's and 35 (one more zero) --> 7 0's

Break down the bases

When working with exponents, it's often helpful to break down the bases to their prime factors Ex: 15^25 --> (3 x 5)^25 8^12 --> (2 x 2 x 2)^12 --> (2^3)^12 --> 2^36

Can an instance of division that has a quotient of 0 have a remainder?

Yes Ex: 2/7 = 0 Remainder 2 Because 7 goes into 2 zero times but leaves 2 left over when it does

When working with inequalities such as (x^2)y > 0 or (x^4)(y^2)z < 0...

You can always remove the even powered terms to simplify the expression

Combining Equations 4x + y = 9 and 3x - 6y = 0

You can solve by substituting: 1. Isolate a variable they have in common 2. Plug that variable into the other equation It is preferable to isolate a variable with no coefficient. You can also solve by elimination: 1. Stack the two equations atop one another 2. Arrange the variables in the same order 3. Add or subtract the equations to eliminate one of the variables *This only works when combining the equations cancels out one of the variables. For a variable to cancel out, it must have opposite coefficients. You can multiply one or both of the equations by some value to give either variable opposite coefficients. You can also try to solve by "chunks" to see if you can get one of the quantities you're comparing directly.

Manipulating "plug in" questions (the more difficult way to solve)

You can: 1. ADD/SUBTRACT anything you want from both quantities 2. MULTIPLY/DIVIDE both quantities by a POSITIVE value 3. SQUARE or SQUARE ROOT both quantities, if they are POSITIVE Cannot multiply/divide by negative values or square/square root by negative values

RARE: Functions & Sequences: Finding Patterns Ex: For every integer n from 1 to 20 inclusive, the nth term of a certain sequence is defined as -1(n^1)(1/4^n). If z is the sum of the first 10 terms in the sequence, then z should have which of the following values?

You have to recognize a pattern within a sequence. n = 1, = 1/4 n = 2, n = -1/16 n = 3, n = 1/64 n = 4, n = -1/256 So answer is 0 < z < 1/4 You may have to look for a pattern in more than just the sequence

If a variable has a fraction for a coefficient when you're solving an equation.... Ex: (2/3)n = 4

You may find it easier to multiply by its reciprocal than to use the denominator trick Just make sure before you do this that the variable stands alone (i.e., nothing other than the coefficient)

Determining if a number less than 100 is prime

You only have to test 2, 3, 5, and 7 as factors

RARE Root Rule: Conjugation 5 / 2 + Root(3)

[5 / 2 + Root(3)] x [2- Root(3) / 2 - Root(3)] and then do difference of squares

RARE Root Rule: Irrational Fractions 8 / Root(n)

[8 / Root(n)] x [Root(n) / (Root(n)] = [8Root(n)]/n

Exponent Rule: Exponent Equations: Make Bases the Same, Set the Exponents Equal x^(a+4) = x^(8-3a) 8^2 = 2^x

a + 4 = 8 - 3a (2^3)^2 = 2^x --> 6 = x

Sequences

a set of number whose order is governed by a rule or formula. Similarly to formulas, they have: The Header: S1, S2, S3, Sn... The Formula: Sn = 2n-7 The Input: S6 The input and formula tell you what to plug in and where to plug it. Ex: For all positive integers n, the sequence S1, S2, S3...Sn, is such that Sn = 2n-7. What is the value of S6? Sometimes they have constraints.

One number is divisible by another if...

all of the prime factors of the divisor are contained in the number being divided for example, 20 is not divisible by 6 because 6 = 2 x 3, but only a 2 is contained in the prime factors of 20

Exponent(ial) Equation

any equation in which a variable occur in the exponent Ex: 2^(2x-1) = 16 To solve: you need to make the bases the same, and then you can set their exponents equal to solve for x Ex: 2^(2x-1) = 8, x must equal 2, since 8=2^3 2^(2x-1) = 2^3 2x - 1 = 3 2x = 4, so x = 2 In most cases, you will need to break down one or both of your bases to make them the same

Exponent(ial) Inequality

any inequality in which a variable occurs in the exponent - they work like exponential equations Ex: n>0 and 625^n < 5^(1-n) 5^4n < 5^1-n 4n = 1 - n 5n = 1 n = 1/5

Power of Zero

any term raised to 0 always equals 1 (except 0)

The Power of One

any term raised to the power of 1 is equal to itself *If you're working with exponents and a term doesn't have one, it can make it easier to write it as 5 --> 5^1

Square

any term raised to the second power

Cube

any term raised to the third power

Coefficient

any term situated before the base Ex: 2y^3 Coefficient: 2