Algebra

Compound Base

Similar to distributing numerator across a fraction, it can be distributed across a product: 10^3 = (2 x 5)^3 = (2)^3 x (5)^3 = 8 x 125 = 1000 Works with variables too: (3x)^4 = (3)^4 x (x)^4 = 81x^4

Multiple Equations

Solve by Substitution or Combination

Three Special Products

Special Product #1: x^2 - y^2 = (x + y)(x - y) Special Product #2: x^2 + 2xy + y^2 = (x + y)(x + y) = (x + y)^2 Special Product #3: x^2 - 2xy + y^2 = (x - y)(x - y) = (x - y)^2

Linear Equations

all variables have an exponent of 1

Equations

contain an equal sign any change made to one side must also be made to the (entire) other side also

√ 121

11

√ 144

12

√ 169

13

√ 196

14

√ 225

15

√ 256

16

√ 4

2

√ 5

2.25

√ 400

20

√ 625

25

√ 9

3

√ 900

30

FOIL

(x + 6) (x +2) First: x * x Outer: x * 2 Inner: 6 * x Last: 6 * 2 x^2 + 2x + 6x + 12= x^2 +8x +12

Absolute Value Equations

- refers to the positive value of the expression - generally have two solutions (that could make equation true) - because value of expression inside AV brackets could be +/- Solve with following steps Step 1: Isolate the expression within the AV brackets Step 2: Once in the from of |x| = a with a > 0, remove the AV brackets and solve the equation for 2 different cases (positive & negative) *Step 3: Check to see whether each solution is valid by inputting into the original equation and verifying both sides are equal*

Base of 0 or 1 (Exponential Expression)

0 raised to any power equals 0 1 raised to any power equals 1

√ 1

1

Factoring Quadratic Equations

1) Move all terms to the left side of the equation in form of ax^2+bx+c = 0 2) Factor where the two terms have a product equal to and a sum equal to b 3) Rewrite into from (x + ?)(x + ?) 4) Since this product equals 0, one or both of the factors must equal 0

√ 2

1.4

√ 3

1.7

√ 100

10

√ 16

4

√ 25

5

√ 36

6

√ 49

7

√ 64

8

√ 81

9

Strange Symbol Formulas

Arbitrary Symbol that defines a certain procedure Interpret the formula step-by-step and understand what it means to calculate a solution for the formula with the actual numbers.

True/False: You can seperate/combine the sum or difference of two roots

FALSE

True/False: x^3 + x^2 = x^5

FALSE

Variable Base (Exponential Expression)

Can be raised to an exponent and behave the same as numbers.

Sequence Formulas

Collection of numbers in a set order...every sequence is defined by a rule which you can use to find the values of the term.

Recursive Sequence

Defines each term relative to other terms

3w^2 = 6w , 36/b = b - 5

Disguised Quadratics

Subtraction of Expressions

Each term in the subtracted part must have its sign reversed

Same Base or Same Exponent

Eliminate the bases or exponents and rewrite the remainder of an equation

True/False: absolute value signs are not equivalent to parentheses

False

True/False: Zero in the Denominator provides a solution

False - Undefined

True/False: Multiplying or dividing an inequality by a negative number does not change the direction of the inequality symbol.

False...inequality flips

Sequence Problems: Alternate Method

For simple linear sequences, instead of finding rule for sequence, consider following reasoning: If each number in a sequence is three more than the previous number, and the sixth number is 32, what is the 100th number? From the 6th to the 100th term, there are 94 "jumps" of 3. 94x3=282 increase from 6th to 100th term. 32 + 282 = 314.

Plug-In Formulas

Formula problems that provide a formula and asks to solve for one of the variable in the formula by plugging in the given values for the other variables. Typically have to do some rearrangement after plugging in the numbers to isolate the desired unknown.

Sequences - Look for Patterns instead of Rules

If Sn = 3^n, what is the units digit of S65? Key is too identify what the pattern is for powers of 3: 3^1 = 3 3^2 = 9 3^3 = 27 3^4 = 81 3^5 = 243 3^6 = 729 3^7 = 2187 3^8 = 6561 Patter for powers of 3: 3, 9, 7, 1, [repeating].... when n=multiple of 4=1. Use that for anchor points -> 65 is 1 more than 64 (closest multiple of 4), unit digit of S65 will be 3, which always follows 1 in the pattern.

Fraction Bars as Grouping Symbols

In any expression with a fraction bar, pretend that there are parentheses around the numerator and denominator of the fraction

Solving One-Variable Equations

Isolate the variable on one side of the equation

Formulas with Unspecified Amounts

No real values -> only indicate how the value of a variable has changed. e.g. You must express the new cost in terms of the original cost.

Fractional Exponents

Numerator tells you what power to raise the base; Denominator tells you what root to take

One-Solution Quadratics

Perfect square x^2 + 8x + 16 = 0 (x + 4)(x + 4) = 0 (x + 4)^2 = 0

Fractional Base (Exponential Expression)

Positive proper fraction: as the exponent increases, the value of the expression decreases! Can distribute exponent across numerator & denominator first

Negative Base (Exponential Expression)

Raised to odd exponents -> NEGATIVE Raised to even exponents -> POSITIVE Pay particular attention to PEMDAS (Unless negative sign is inside parentheses, the exponent does not distribute)

Simplifying Roots of Imperfect Squares

Rewrite imperfect square (√ 52) as a product of primes: √ 2 x 2 x 13 = √ 4 x 13 = 2√ 13

True/False: You cannot multiply or divide an inequality by a variable, unless you know the sign of the number that the variable stands for.

True

True/False: Adding or subtracting a constant or variable expression on both sides of an inequality does not change the direction of the inequality symbol.

True

True/False: If you have a quadratic equal to 0, and you can factor an x out of the expression, then x=0 is a solution.

True

True/False: Odd Exponents keep the sign of the base

True

Expressions

nothing can be done to change the value

Anything Raised to the Zero Power equals ____

one

Quadratic Equations

one unknown and two defining components: 1) variable raised to the second power 2) variable raised to the first power Generally have 2 solutions (due to even exponent)

If an equation contains a square root....

only use the positive root (regardless of odd/even roots)

Negative Exponents

something with a negative exponent is just "one over" that same thing with a positive exponent. y^-3 = 1 / y^3

y^6 / y^2

y^(6-2)

(z^2)(^3)

z^(2*3)

z^2 * z^3

z^(2+3)

√ x^2

|x| Even exponents hide the sign of the base

√ x * y

√ x * √ y

√ x / z

√ x / √ z