Algebra Rules

Product of powers

$a^m \cdot a^n = a^{m+n}$

$2^3 \cdot 2^2 = 2^5 = 32$

Quotient of powers

$a^m / a^n = a^{m-n}$

$x^5 / x^2 = x^3$

Power of a power

$(a^m)^n = a^{mn}$

$(x^2)^3 = x^6$

Power of a product

$(ab)^n = a^n b^n$

$(2x)^3 = 8x^3$

Negative exponent

$a^{-n} = 1/a^n$

$2^{-3} = 1/8$

Zero exponent

$a^0 = 1\ (a \ne 0)$

$7^0 = 1$

Fractional exponent

$a^{1/n} = \sqrt[n]{a}$

$9^{1/2} = 3$

Distributive

$a(b+c) = ab + ac$

$3(x+2) = 3x + 6$

Square of a sum

$(a+b)^2 = a^2 + 2ab + b^2$

$(x+3)^2 = x^2 + 6x + 9$

Square of a difference

$(a-b)^2 = a^2 - 2ab + b^2$

$(x-2)^2 = x^2 - 4x + 4$

Difference of squares

$a^2 - b^2 = (a+b)(a-b)$

$x^2 - 9 = (x+3)(x-3)$

Common factor

$ab + ac = a(b+c)$

$6x^2 + 4x = 2x(3x + 2)$

Quadratic formula

$x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

$x^2 - 5x + 6 = 0 \Rightarrow x = 2,\,3$

Discriminant

$\Delta = b^2 - 4ac$

$\Delta > 0$: two real roots; $\Delta = 0$: one; $\Delta < 0$: none real

Sum / product of roots

$x_1 + x_2 = -b/a,\ \ x_1 x_2 = c/a$

For $x^2 - 5x + 6$: sum $=5$, product $=6$