A surd is a root that does not come out evenly, like 5. They are hard to work with on the bottom of a fraction, so the main job is to get rid of them there.
Two moves cover almost everything
1. Rationalising — multiply by the conjugate. The conjugate of a+b is a−b (just flip the middle sign). Multiply top and bottom by it; the bottom becomes a2−b, a plain whole number, because (a+b)(a−b)=a2−b.
Example: 3+51×3−53−5=9−53−5=43−5.
2. Nested surds — rewrite as a perfect square. For a+bc, find x and y so that x + y = a and 2xy=bc; then it equals x+y. Example: 7+43=(2+3)2, so 7+43=2+3.
For 4+71, the conjugate is 4−7, and the bottom becomes 42−7 = .
📝Practice Questions
Q1Rationalise: 1/(√5 − 2). The denominator becomes:
Q2Simplify √(9 + 4√5):
Q3Which equals √50?
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Real exam questions — Surds & Rationalization
2 question types · 6 solved examples from real SSC papers
Two moves cover most surd questions: multiply by the conjugate, or rewrite a nested surd as a perfect square.
How to solve this type
A surd like √5 sitting in the denominator is hard to work with, so we remove it. Take the conjugate of the bottom — the same two terms with the middle sign flipped (a−√b for a+√b). Multiply both the top and the bottom by it; the bottom becomes a²−b, a plain whole number with no root left. Then simplify the top over that whole number.
