Surds

1. Simplify surds like √48 by taking out square factors

   Look for the biggest square number hiding inside: 48 = 16 × 3, so $\sqrt{48}$ = $\sqrt{16}$ × $\sqrt{3}$ = $4\sqrt{3}$. Working with the largest square factor gets you there in one step, but pulling out smaller squares repeatedly works too.

2. Multiply and divide surds, and expand brackets containing surds

   Surds multiply and divide just like whole numbers under one root: $\sqrt{2}$ × $\sqrt{8}$ = $\sqrt{16}$ = 4. Expanding brackets follows normal algebra - $1+\sqrt{3}2-\sqrt{3}$ = 2 − $\sqrt{3}$ + $2\sqrt{3}$ − 3 = −1 + $\sqrt{3}$. Collect the rational parts and the surd parts separately.

3. Rationalise denominators of the form a/√b

   A tidy answer never leaves a surd on the bottom. Multiply top and bottom by the surd: $\frac{5}{\sqrt{3}}$ = $\frac{5\sqrt{3}}{3}$, because $\sqrt{3}$ × $\sqrt{3}$ = 3. The value is unchanged - you have multiplied by 1.

4. Rationalise denominators of the form a/(b + √c)

   Multiply top and bottom by the CONJUGATE - the same expression with the sign flipped. For $\frac{2}{3-\sqrt{5}}$, use $3+\sqrt{5}$: the bottom becomes 9 − 5 = 4 because the middle terms cancel, leaving $\frac{6+2\sqrt{5}}{4}$ = $\frac{3+\sqrt{5}}{2}$.

5. Give exact answers in surd form, and know when exact form is asked for

   When a question says "give your answer in exact form" or "in the form $a\sqrt{b}$", a rounded decimal scores nothing - $2\sqrt{3}$ is exact, 3.46 is not. Keep surds through your working rather than reaching for the calculator, and simplify at the end.

More Number topics

• Fractions
• Percentages
• Standard Form
• Indices
• Bounds & Rounding
• Factors & Multiples

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