--- title: "Laws of Exponents" description: "Apply integer-exponent rules for nonzero bases: add exponents when multiplying matching-base powers, subtract when dividing, multiply when raising a power to a power, and interpret zero and negative e" canonical: https://lightmysky.com/learn/mathematics/laws-of-exponents-mt_UjNba32kkz source: https://lightmysky.com/learn/mathematics/laws-of-exponents-mt_UjNba32kkz.md retrieved: 2026-11-12 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # Laws of Exponents Apply integer-exponent rules for nonzero bases: add exponents when multiplying matching-base powers, subtract when dividing, multiply when raising a power to a power, 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 nonzero base, and powers raised to powers (e.g., 5³ × 5⁴ = 5⁷, 2⁸ ÷ 2³ = 2⁵, (3²)⁴ = 3⁸) - 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 You have used powers to count repeated factors. In 3² = 3 × 3, the base is 3 and the exponent 2 counts factors. With matching bases, keep the base and add exponents: 3² × 3⁴ = 3⁶. Do not multiply the matching bases. A letter such as n, x, y or p can be a base too. We write the product rule as aᵐ × aⁿ = aᵐ⁺ⁿ, where a is the base and m and n are exponents. *(drawing: The product has six factors of 3. Adding the exponents counts those factors.)* For division with matching nonzero bases, keep the base and subtract exponents: aᵐ ÷ aⁿ = aᵐ⁻ⁿ. For example, 8⁵ ÷ 8² = 8³ because 5 − 2 = 3. Cancelling two factors of 8 from both numerator and denominator leaves three factors of 8. Check the bases and operation first. The product and quotient rules do not apply to unlike bases or to addition. A power raised to a power repeats the whole inner power. For example, (2²)⁴ has four factors of 2². Each contains two factors of 2, so there are 2 × 4 = 8 factors of 2 altogether: (2²)⁴ = 2⁸. For (aᵐ)ⁿ, multiply m and n to get the new exponent. Keep the base a. Any nonzero base raised to zero equals 1: a⁰ = 1 for a ≠ 0. The sign ≠ means 'is not equal to'. For example, 9³ ÷ 9³ = 1, and subtracting exponents gives 9⁰. The rule does not apply to 0⁰. Check the whole base inside any brackets. In 5 × 2⁰, only 2 is raised to zero, so the result is 5 × 1 = 5. For a nonzero base a and a positive integer n, a⁻ⁿ = 1/aⁿ. A negative exponent means the reciprocal of the positive power. For example, 4⁻² = 1/4² = 1/16. It does not put a minus sign in front of the answer. To compare negative powers of positive bases, evaluate their fractions. Among fractions 1/d with positive denominators, the larger denominator gives the smaller fraction. For a nonzero base a, combine integer exponents from left to right: add for × and subtract for ÷. Adding −2 means subtracting 2. Subtracting −2 means adding 2: 6² ÷ 6⁻² = 6⁴. For a negative result −k, use a⁻ᵏ = 1/aᵏ, where k is a positive integer. Multiply k factors of a to find the denominator. For example, 6³ × 6⁻² ÷ 6² gives 3 + (−2) − 2 = −1, so the result is 6⁻¹ = 1/6. **Recap.** For a nonzero base, add exponents to multiply, subtract to divide, and multiply for a power of a power. A zero exponent gives 1; a negative exponent gives a reciprocal. *Original lesson, practice questions and flashcards written for this dataset.* ## 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) ## Optional connections - [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)