Exponents — Formula Sheet

x^n

x \times x \times x \times \dots \text{ n times } = x^n

x^{-n} = 1/x^n

a^m \times a^n = a^{m+n}

a^m / a^n = a^{m-n}, if m > n

a^m / a^n = 1/a^{n-m}, if n > m

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

(a \times b)^n = a^n \times b^n

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

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

a^0 = 1

(-a)^m = a^m, if m is even

(-a)^m = -a^m, if m is odd

a^{-n} = 1/a^n and a^n = 1/a^{-n}.

When powers have the same base, multiplication adds the exponents: a^m × a^n = a^(m+n).

For example, 3^3 × 3^5 = 3^8.

If m>n, a^m / a^n = a^(m-n).

If n>m, a^m / a^n = 1/a^(n-m).

When a power is raised to another power, the exponents are multiplied: (a^m)^n = a^(mn).

For example, (3^5)^2 = 3^10.

For any non-zero rational number a, a^-n = 1/a^n and a^n = 1/a^-n.