--- title: "Surds: Simplifying and Rationalising" description: "Simplify surds by taking out square factors, add and multiply them, and clear a surd from the bottom of a fraction." canonical: https://lightmysky.com/learn/mathematics/surds-simplifying-and-rationalising-mt_cqr0crUUXS source: https://lightmysky.com/learn/mathematics/surds-simplifying-and-rationalising-mt_cqr0crUUXS.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`. # Surds: Simplifying and Rationalising Simplify surds by taking out square factors, add and multiply them, and clear a surd from the bottom of a fraction. Subject: Mathematics · Area: Number Representation & Place Value · Ages 15 to 16 Page: https://lightmysky.com/learn/mathematics/surds-simplifying-and-rationalising-mt_cqr0crUUXS ## Ready when they can - Simplify √50 to 5√2 by finding the largest square factor - Expand a bracket containing surds and collect like terms - Rationalise a denominator such as 6/√2 ## Lesson: Tidying messy roots A surd is a root you leave exact, and simplifying means hunting the biggest square factor inside. 50 splits into 25 times 2, and root 25 is 5, so root 50 becomes 5 root 2. The same hunt turns root 72 into 6 root 2. **Example.** Brackets with surds expand like ordinary brackets, then only matching roots merge. Root 2 times (root 8 + root 2): root 2 times root 8 is root 16 = 4, and root 2 times root 2 is 2, giving 6. And (root 5 + 1) times (root 5 - 1) is 5 - 1 = 4. Examiners refuse roots on the bottom of a fraction, so you evict them by multiplying top and bottom by the root. 6 over root 2: multiply by root 2 over root 2 to get 6 root 2 over 2, which simplifies to 3 root 2. The root migrates upstairs where it belongs. **Tip.** Only matching roots combine in a sum. 2 root 3 plus 5 root 3 is 7 root 3, but root 2 plus root 3 cannot merge. Simplify every root first, then look for matches. **Recap.** Split out the biggest square, merge only matches, multiply away bottom roots. ## Practice 14 questions on this page, each with its working shown. ## Needs first - [Number Sets & Infinity](https://lightmysky.com/learn/mathematics/number-sets-and-infinity-mt_b4lbTOJYwI) - [Fractional Indices and Roots](https://lightmysky.com/learn/mathematics/fractional-indices-and-roots-mt_fdK-nbQcBu) ## Opens up - [The Quadratic Formula](https://lightmysky.com/learn/mathematics/the-quadratic-formula-mt_C-4brnRYHY) - [Proof by Contradiction and Counterexample](https://lightmysky.com/learn/mathematics/proof-by-contradiction-and-counterexample-mt_pbFTv4v6k8) - [Exact Values of Sine, Cosine and Tangent](https://lightmysky.com/learn/mathematics/exact-values-of-sine-cosine-and-tangent-mt_SNJEbXYOrR)