Introduction to Exponents

Exponent Notation

Base and exponent (or power): $a^n = \underbrace{a \times a \times \cdots \times a}_\text{n times}$

Example: $5^3 = 5 \times 5 \times 5 = 125$

Read as: "5 to the third power" or "5 cubed"

Special Cases

Any number to the first power equals itself: $a^1 = a$

Any number (except 0) to the zero power equals 1: $a^0 = 1$

Perfect Squares

Numbers that are squares of whole numbers: $1, 4, 9, 16, 25, 36, 49, 64, 81, 100, ...$

$4 = 2^2, \quad 9 = 3^2, \quad 16 = 4^2$

Perfect Cubes

$1, 8, 27, 64, 125, ...$

$8 = 2^3, \quad 27 = 3^3, \quad 64 = 4^3$

Product Rule

When multiplying with the same base, add the exponents: $a^m \times a^n = a^{m+n}$

Example: $2^3 \times 2^4 = 2^{3+4} = 2^7 = 128$

Quotient Rule

When dividing with the same base, subtract the exponents: $\frac{a^m}{a^n} = a^{m-n}$

Example: $\frac{5^6}{5^2} = 5^{6-2} = 5^4 = 625$

Power Rule

When raising a power to a power, multiply the exponents: $(a^m)^n = a^{mn}$

Example: $(3^2)^3 = 3^{2 \times 3} = 3^6 = 729$