---
title: "Laws of Exponents"
description: "Apply the rules of integer exponents to write equivalent expressions: multiply powers by adding indices, divide by subtracting, raise a power to a power by multiplying, and interpret zero and negative"
canonical: https://lightmysky.com/learn/mathematics/laws-of-exponents-mt_UjNba32kkz
source: https://lightmysky.com/learn/mathematics/laws-of-exponents-mt_UjNba32kkz.md
retrieved: 2026-10-19
---

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# Laws of Exponents

Apply the rules of integer exponents to write equivalent expressions: multiply powers by adding indices, divide by subtracting, raise a power to a power by multiplying, and interpret zero and negative exponents (3² × 3⁻⁵ = 3⁻³ = 1/27)

Subject: Mathematics · Area: Algebra · Ages 12 to 14
Page: https://lightmysky.com/learn/mathematics/laws-of-exponents-mt_UjNba32kkz

## Ready when they can

- Simplify products and quotients of powers with the same base (e.g., 5³ × 5⁴ = 5⁷, 2⁸ ÷ 2³ = 2⁵)
- Evaluate expressions with zero and negative exponents (e.g., 7⁰ = 1, 2⁻³ = 1/8)
- Spot and correct a common exponent error, such as claiming 3² × 3³ = 9⁵

## Lesson: The Laws of Exponents

An exponent tells you how many times to multiply a number by itself. When two powers share the same base, like 2³ and 2⁴, you can multiply them by adding the exponents: 2³ × 2⁴ = 2⁷. Here's why: 2³ is three 2s multiplied together, and 2⁴ is four more 2s. Put them together and you get seven 2s multiplied together, which is 2⁷.

*(drawing: Three 2s times four 2s makes seven 2s: 2³ × 2⁴ = 2⁷.)*

**Example.** Farah simplifies 5⁶ ÷ 5². She keeps the base 5 and subtracts the exponents: 6 − 2 = 4. So 5⁶ ÷ 5² = 5⁴. Dividing cancels matching 5s from the top and bottom, leaving four 5s multiplied together.

There's also a rule for a power raised to another power, like (2³)². This means 2³ multiplied by itself twice, so you multiply the exponents instead of adding them: 3 × 2 = 6. That gives (2³)² = 2⁶.

**Tip.** Zero and negative exponents follow the pattern from dividing. Since 2³ ÷ 2³ = 2⁰, and a number divided by itself is always 1, you get 2⁰ = 1. Go one step further, like 2² ÷ 2⁵ = 2⁻³, and a negative exponent means flip it into a fraction: 2⁻³ = 1/2³ = 1/8. Watch out for a common mistake: 3² × 3³ is 3⁵, not 9⁵. The base stays 3; only the exponents add.

**Recap.** Same base: add exponents to multiply, subtract to divide, multiply for a power of a power, and remember zero gives 1 while negative flips to a fraction.

## Practice

14 questions on this page, each with its working shown.

## Needs first

- [Integer powers and roots](https://lightmysky.com/learn/mathematics/integer-powers-and-roots-mt_hCVPYlF-7Y)
- [Patterns with Powers of Ten](https://lightmysky.com/learn/mathematics/patterns-with-powers-of-ten-mt_HLUqHJ9Y7n)

## Opens up

- [Zero and Negative Indices](https://lightmysky.com/learn/mathematics/zero-and-negative-indices-mt_jAtXBSVqkj)