Exponent Rules
Exponent rules let you simplify expressions with powers. The main ones to know:
am · an = am+n
am / an = am-n
(am)n = amn
a0 = 1
a-n = 1 / an
am / an = am-n
(am)n = amn
a0 = 1
a-n = 1 / an
Example: 2³ × 2⁴ = 2⁷ = 128.
Linear Equations
A linear equation in one variable has the form ax + b = 0. Solving for x gives:
x = -b / a
For two variables (a straight line on a graph): y = mx + c, where m is the slope and c is the y-intercept.
Quadratic Formula
For any quadratic equation ax² + bx + c = 0, the solutions are:
x = (-b ± √(b² - 4ac)) / 2a
The value b² - 4ac is called the discriminant. If it is positive, there are two real solutions. If zero, one solution. If negative, no real solutions.
Factoring Identities
| Identity | Expanded Form |
|---|---|
| (a + b)² | a² + 2ab + b² |
| (a - b)² | a² - 2ab + b² |
| (a + b)(a - b) | a² - b² |
| (a + b)³ | a³ + 3a²b + 3ab² + b³ |
| (a - b)³ | a³ - 3a²b + 3ab² - b³ |
Laws of Logarithms
log(ab) = log(a) + log(b)
log(a/b) = log(a) - log(b)
log(aⁿ) = n · log(a)
logb(x) = ln(x) / ln(b)
log(a/b) = log(a) - log(b)
log(aⁿ) = n · log(a)
logb(x) = ln(x) / ln(b)
Arithmetic & Geometric Sequences
Arithmetic sequence - each term increases by a fixed amount d:
aₙ = a₁ + (n-1)d | Sₙ = n/2 · (a₁ + aₙ)
Geometric sequence - each term is multiplied by a fixed ratio r:
aₙ = a₁ · rn-1 | Sₙ = a₁(1-rⁿ)/(1-r)