--- 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: 2026-11-12 --- > **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`. # 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 You have written roots as fractional indices: √n = n^(1/2). Here √n means the non-negative square root. A surd is an irrational root, such as √5; √9 = 3 is not a surd. For non-negative a and b, √(a × b) = √a × √b. Find the largest square factor: √45 = √(9 × 5) = 3√5. Multiply by the outside coefficient too: 2√45 = 6√5. *(drawing: The square factor is 9. Its square root is 3, which becomes the coefficient of √5.)* Add coefficients only when the simplified root parts match. For example, √20 + √80 = 2√5 + 4√5 = 6√5. In general, a√b + c√b = (a + c)√b. √3 + √7 stays as a sum because the root parts differ. The product rule does not work for addition: √(a + b) need not equal √a + √b. **Example.** For non-negative b and d, (a√b)(c√d) = ac√(bd). Multiply coefficients and numbers under roots, then take out square factors. Expand brackets by multiplying every pair: √3(√12 + √3) = √36 + 3 = 9. Also, (√7 + 2)(√7 - 2) = 7 - 2√7 + 2√7 - 4 = 3. Squaring repeats a bracket: (√2 + 3)^2 = 2 + 3√2 + 3√2 + 9 = 11 + 6√2. Rationalising means rewriting a fraction with a rational denominator. A root in the denominator is valid, but a question may ask for a rational denominator. Multiply the numerator and denominator by the same non-zero factor. For 12/√3, use √3/√3 = 1: 12/√3 = 12√3/3 = 4√3. Multiplying by 1 keeps the value unchanged. *(drawing: Both numerator and denominator are multiplied by √3. The denominator becomes 3.)* If the denominator is c√b, with b positive and c non-zero, multiplying by √b/√b makes it cb. For example, 9/(2√7) = 9√7/14. Multiplying only the denominator changes the fraction's value. Keep the same factor in the numerator too, then reduce any common numerical factors. **Recap.** Take out square factors, add matching simplified root parts, expand every product, and multiply both numerator and denominator to rationalise. *Lesson, practice questions and flashcards written for this dataset.* ## 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)