--- title: "Fractional Indices and Roots" description: "Read x^(1/n) as the nth root and x^(m/n) as a root and a power together, and move between root form and index form." canonical: https://lightmysky.com/learn/mathematics/fractional-indices-and-roots-mt_fdK-nbQcBu source: https://lightmysky.com/learn/mathematics/fractional-indices-and-roots-mt_fdK-nbQcBu.md retrieved: 2026-09-12 --- > **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`. # Fractional Indices and Roots Read x^(1/n) as the nth root and x^(m/n) as a root and a power together, and move between root form and index form. Subject: Mathematics · Area: Number Representation & Place Value · Ages 15 to 16 Page: https://lightmysky.com/learn/mathematics/fractional-indices-and-roots-mt_fdK-nbQcBu ## Ready when they can - Work out 27^(1/3) and 16^(3/4) without a calculator - Rewrite a root as a fractional index and back again - Handle a negative fractional index in one calculation ## Lesson: Roots wearing index outfits x to the power 1/n means the nth root. 27^(1/3) asks which number multiplied by itself three times gives 27, and that number is 3. A power of 1/2 is the square root and 1/5 is the fifth root, so 32^(1/5) = 2. **Example.** For x^(m/n), take the root first, then the power. 16^(3/4): the fourth root of 16 is 2, then 2 cubed is 8. Root first keeps the numbers small. 8^(2/3) works the same way: cube root 2, then squared, giving 4. Roots and indices are the same idea in two outfits, and you can swap them. The cube root of x^5 becomes x^(5/3): inside power on top, root type below. Back the other way, x^(2/5) is the fifth root of x squared. **Tip.** A minus sign only means flip to the reciprocal, and you already know that move. For 27^(-1/3), flip first to 1 over the cube root of 27, which is 1/3. Do the flip, then the root and power, one step at a time. **Recap.** Bottom of the fraction picks the root, top picks the power, minus picks the flip. ## Practice 20 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) - [Zero and Negative Indices](https://lightmysky.com/learn/mathematics/zero-and-negative-indices-mt_jAtXBSVqkj) ## Opens up - [Surds: Simplifying and Rationalising](https://lightmysky.com/learn/mathematics/surds-simplifying-and-rationalising-mt_cqr0crUUXS) - [The Power Rule for Differentiating Polynomials](https://lightmysky.com/learn/mathematics/the-power-rule-for-differentiating-polynomials-mt_l7f4j44bR3)