Basic Identies

(a+b) ² = a ² + 2ab + b ²

(a-b) ² = a ² - 2ab + b ²

a ² + b ² = (a+b) ² - 2ab

a ² + b ² = (a-b) ² + 2ab

a ² - b ² = (a+b)(a-b)

(a+b) ³ = a ³ + 3a²b + 3ab² + b ³

(a-b) ³ = a ³ - 3a²b + 3ab² - b ³

a ³ + b ³ = (a + b)(a² - ab + b²)

a ³ - b ³ = (a - b)(a² + ab + b²)

(a+b)(c+d) = ac + ad + bc + bd

x ² + (a+b)x + ab = (x + a)(x + b)

Powers

x ^a x ^b = x ^{(a + b)}

x ^a y ^a = (xy) ^a

(x ^a) ^b = x ^(ab)

x ^(-a) = 1 / x ^a

x ^{(a - b)} = x ^a / x ^b

Logarithms

y = log_b(x) if and only if x=b ^y

log_b(1) = 0

log_b(b) = 1

log_b(x*y) = log_b(x) + log_b(y)

log_b(x/y) = log_b(x) - log_b(y)

log_b(x ⁿ) = n log_b(x)