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