Surds and radicals
Simplifying surds, writing a surd as a single root and comparing surds, adding, multiplying and dividing surds, rationalizing the denominator and fractional indices. For GCSE Higher (Year 11) and Algebra 1.
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Available exercises
Simplifying surds
Take everything you can out of the root: √72 = 6√2, ∛54 = 3∛2, √(18a³) = 3a√(2a).
Surds as a single root and comparing
Move the number in front of the root under the root (3√2 = √18) and compare the size of surds, for example 2√7 and 3√3.
Adding and subtracting surds
Add and subtract surds: simplify first, then combine like surds — √12 + √27 = 5√3.
Multiplying and dividing surds
Multiply and divide surds, expand brackets and use the identities: √3 · √12 = 6, (√2 + √3)² = 5 + 2√6.
Rationalizing the denominator
Remove the surd from the denominator: 6/√3 = 2√3, 1/(√5 − √2) = (√5 + √2)/3.
Fractional indices
Write a root as a power with a fractional index and the other way round: 8^(2/3) = 4, ∛(a²) = a^(2/3), a^(1/2) · a^(1/3) = a^(5/6).