Exponent Rules
These five rules cover most of what you need to simplify powers without expanding anything by hand. Once they're second nature, exponent problems turn into simple pattern-matching:
am / an = am-n
(am)n = amn
a0 = 1
a-n = 1 / an
For example: 3² × 3³ = 3⁵ = 243 - add the exponents and keep the base as it is.
Solving Linear Equations
Any one-variable linear equation boils down to the same form, ax + b = 0, and solving for x just means isolating it on one side:
Bring in a second variable and you're no longer solving for a single number - you're describing a line: y = mx + c. Here m sets how steep the line is (the slope), and c marks where it crosses the y-axis.
The Quadratic Formula
Any quadratic written as ax² + bx + c = 0 can be solved with this formula, no factoring required:
The real story is under the square root: b² - 4ac, known as the discriminant, tells you what kind of answer you're getting before you finish the calculation. A positive value means two distinct real solutions, zero means exactly one repeated solution, and a negative value means there's no real solution at all. If you'd rather skip the arithmetic, SolverCalc doubles as a calc solver for this - enter a, b, and c and it hands back both roots plus the discriminant.
Common Factoring Identities
Recognizing these shapes lets you expand or factor an expression in one move instead of multiplying every term out by hand:
| 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 |
Logarithm Rules
Logs turn multiplication into addition and division into subtraction, which is exactly why they're so handy for untangling messy expressions:
log(a/b) = log(a) - log(b)
log(aⁿ) = n · log(a)
logb(x) = ln(x) / ln(b)
Arithmetic and Geometric Sequences
Arithmetic sequence - every term is the one before it plus a fixed amount d:
Geometric sequence - every term is the one before it times a fixed ratio r: