🔢 Surd Secrets Unlocked! Simplify & Rationalize with Confidence

Surds show up all over the O-Level syllabus— from the length of a triangle’s hypotenuse to exact values of geometric ratios. **Mastering surds is vital** because exam setters love testing whether you can keep answers exact rather than rounded. Once you know how to simplify and rationalize, tricky roots become neat, mark-earning expressions.

Main Concept: Simplify First, Rationalize Second

A *surd* is an irrational root that cannot be expressed as a terminating or recurring decimal, e.g.

\sqrt{3}

2\sqrt{5}

. **Key ideas:** (1) Break the number under the root into factors containing perfect squares, (2) Pull perfect squares outside the root, (3) When a denominator contains a surd, multiply by a form of

1

that eliminates the surd.

Worked Example: Tidy Up the Fraction

Problem: Simplify and rationalize

\displaystyle \frac{5\sqrt{12}}{3\sqrt{3}}

. Solution Steps: 1. **Simplify the numerator:**

\sqrt{12}=\sqrt{4\times3}=2\sqrt{3}

, so

5\sqrt{12}=5\times2\sqrt{3}=10\sqrt{3}

. 2. **Cancel common factors:**

\frac{10\sqrt{3}}{3\sqrt{3}} = \frac{10}{3}

(because

\sqrt{3}

divides out). 3. **Check for surds in the denominator:** None remain, so no further rationalization is required. Therefore, the fully simplified result is

\frac{10}{3}.

Keep practising these steps on past-paper questions; with enough repetition, **surds will switch from scary to straightforward!**