Index Laws — The Basics

Read through these examples, then test yourself with questions.

1. What is a power?

A power (also called an index or exponent) is a shorthand for repeated multiplication.

2⁵ = 2 × 2 × 2 × 2 × 2 = 32

The small number (5) tells you how many times to multiply the base (2) by itself. You read it as "two to the power of five".

3⁴ = 3 × 3 × 3 × 3 = 81

When you multiply powers with the same base, you add the powers. Here is why:

2² × 2³ = (2×2) × (2×2×2) = 2×2×2×2×2 = 2⁵ = 32

Count the twos: there are 2 + 3 = 5 of them. So 2² × 2³ = 2⁽²⁺³⁾ = 2⁵.

a^m × aⁿ = a^m+n

In the quiz, you will write out the full expansion before computing the answer.

When you divide powers with the same base, you subtract the powers:

2⁵ ÷ 2² = (2×2×2×2×2) ÷ (2×2) = 2×2×2 = 2³ = 8

Five twos on top, two twos cancel from the bottom, leaving 5 − 2 = 3 twos.

a^m ÷ aⁿ = a^m−n

When you raise a power to another power, you multiply the powers:

(2³)² = 2³ × 2³ = (2×2×2) × (2×2×2) = 2⁶ = 64

(a^m)ⁿ = a^m×n

Any number to the power 0 equals 1. Always.

5⁰ = 1 | 100⁰ = 1 | 7⁰ = 1

For expanded form questions, use × or x or * between numbers. For example:

3×3×3 or 3x3x3 or 3*3*3

All three are accepted. Spaces don't matter.