Math Wizard
Math Wizard

Simplifying Surds

National 5 Expressions and Formulae

A surd is a root that cannot be written exactly as a whole number or a fraction. \(\sqrt{9}\) is not a surd, because it is exactly 3. \(\sqrt{3}\) is a surd, because its decimal expansion never terminates and never repeats, so writing 1.732 instead loses accuracy.

In National 5 you are expected to leave answers in exact surd form rather than rounding. Simplifying a surd means pulling out the largest square number hiding inside it, so \(\sqrt{48}\) becomes \(4\sqrt{3}\). Once two surds have the same root, they can be added and subtracted like terms in algebra.

Surds appear in Paper 1, the non-calculator paper, and they also turn up inside Pythagoras and trigonometry answers later in the course. Getting fluent here saves marks across several topics.

Key formulas

\[\sqrt{a} \times \sqrt{b} = \sqrt{ab} \qquad \frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}} \qquad \left(\sqrt{a}\right)^2 = a \qquad \frac{a}{\sqrt{b}} = \frac{a\sqrt{b}}{b}\]

Mistakes students actually make

  • Splitting a sum under the root. \(\sqrt{a+b}\) is not \(\sqrt{a}+\sqrt{b}\). Check it: \(\sqrt{9+16}=5\), but \(\sqrt{9}+\sqrt{16}=7\). The multiplication law works; the addition law does not exist.
  • Stopping too early. Writing \(2\sqrt{12}\) and moving on. There is still a square number inside, so it simplifies again to \(4\sqrt{3}\). Always ask whether the number under the root has a square factor left.
  • Leaving a surd on the bottom. An answer of \(\frac{6}{\sqrt{3}}\) is not in its simplest form and will lose the final mark. Rationalise it to \(2\sqrt{3}\).
  • Squaring only part of the term. \(\left(3\sqrt{2}\right)^2\) is \(9 \times 2 = 18\), not 6. The 3 gets squared as well as the root.

Worked example

\[\begin{aligned} &\text{Simplify } \sqrt{45} + \sqrt{20} - \sqrt{5}. \\[4pt] \sqrt{45} &= \sqrt{9 \times 5} = 3\sqrt{5} \\ \sqrt{20} &= \sqrt{4 \times 5} = 2\sqrt{5} \\[4pt] \text{So } \sqrt{45} + \sqrt{20} - \sqrt{5} &= 3\sqrt{5} + 2\sqrt{5} - \sqrt{5} \\ &= 4\sqrt{5} \end{aligned}\]

Practice questions

Question 1 - 2 marks
\[\text{Simplify } \sqrt{48}.\]
\[\begin{aligned}\sqrt{48} &= \sqrt{16 \times 3} \\ &= \sqrt{16} \times \sqrt{3} \\ &= 4\sqrt{3}\end{aligned}\]

Common mistake: Choosing 4 x 12 instead of 16 x 3 and stopping at 2 times root 12.

Question 2 - 3 marks
\[\text{Simplify } \sqrt{72} + \sqrt{32}.\]
\[\begin{aligned}\sqrt{72} &= \sqrt{36 \times 2} = 6\sqrt{2} \\ \sqrt{32} &= \sqrt{16 \times 2} = 4\sqrt{2} \\ \sqrt{72}+\sqrt{32} &= 6\sqrt{2} + 4\sqrt{2} = 10\sqrt{2}\end{aligned}\]

Common mistake: Adding the numbers under the roots to get root 104.

Question 3 - 2 marks
\[\text{Express } \sqrt{5} \times \sqrt{15} \text{ in its simplest form.}\]
\[\begin{aligned}\sqrt{5} \times \sqrt{15} &= \sqrt{75} \\ &= \sqrt{25 \times 3} \\ &= 5\sqrt{3}\end{aligned}\]

Common mistake: Reaching root 75 and not simplifying further.

Question 4 - 2 marks
\[\text{Simplify } \left(3\sqrt{2}\right)^2.\]
\[\begin{aligned}\left(3\sqrt{2}\right)^2 &= 3^2 \times \left(\sqrt{2}\right)^2 \\ &= 9 \times 2 \\ &= 18\end{aligned}\]

Common mistake: Squaring only the root and writing 6.

Question 5 - 2 marks
\[\text{Express } \dfrac{6}{\sqrt{3}} \text{ with a rational denominator.}\]
\[\begin{aligned}\frac{6}{\sqrt{3}} &= \frac{6}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} \\ &= \frac{6\sqrt{3}}{3} \\ &= 2\sqrt{3}\end{aligned}\]

Common mistake: Forgetting to cancel the 6 over the 3 at the end.

Question 6 - 3 marks
\[\text{Express } \dfrac{10}{2\sqrt{5}} \text{ with a rational denominator.}\]
\[\begin{aligned}\frac{10}{2\sqrt{5}} &= \frac{5}{\sqrt{5}} \\ &= \frac{5}{\sqrt{5}} \times \frac{\sqrt{5}}{\sqrt{5}} \\ &= \frac{5\sqrt{5}}{5} = \sqrt{5}\end{aligned}\]

Common mistake: Multiplying by the whole denominator 2 root 5 rather than just root 5.

Question 7 - 3 marks
\[\text{Simplify } \sqrt{27} - \sqrt{12}.\]
\[\begin{aligned}\sqrt{27} &= \sqrt{9 \times 3} = 3\sqrt{3} \\ \sqrt{12} &= \sqrt{4 \times 3} = 2\sqrt{3} \\ \sqrt{27}-\sqrt{12} &= 3\sqrt{3} - 2\sqrt{3} = \sqrt{3}\end{aligned}\]

Common mistake: Writing the final answer as 1 root 3 is fine, but subtracting under the root to get root 15 is not.

Question 8 - 3 marks
\[\text{Expand and simplify } \sqrt{3}\left(\sqrt{6} + \sqrt{3}\right).\]
\[\begin{aligned}\sqrt{3}\left(\sqrt{6}+\sqrt{3}\right) &= \sqrt{18} + \sqrt{9} \\ &= \sqrt{9 \times 2} + 3 \\ &= 3\sqrt{2} + 3\end{aligned}\]

Common mistake: Leaving the answer as root 18 plus 3 without simplifying root 18.

Question 9 - 3 marks
\[\text{Evaluate } \left(\sqrt{7} + 2\right)\left(\sqrt{7} - 2\right).\]
\[\begin{aligned}\left(\sqrt{7}+2\right)\left(\sqrt{7}-2\right) &= \left(\sqrt{7}\right)^2 - 2\sqrt{7} + 2\sqrt{7} - 4 \\ &= 7 - 4 \\ &= 3\end{aligned}\]

Common mistake: Missing that the two middle terms cancel, and leaving root 7 in the answer.

Question 10 - 2 marks
\[\text{Simplify } \dfrac{\sqrt{50}}{\sqrt{2}}.\]
\[\begin{aligned}\frac{\sqrt{50}}{\sqrt{2}} &= \sqrt{\frac{50}{2}} \\ &= \sqrt{25} \\ &= 5\end{aligned}\]

Common mistake: Simplifying root 50 to 5 root 2 first is fine, but dividing the roots is quicker.

Want 1-to-1 help with Simplifying Surds? Message us on WhatsApp and we'll tailor a session around it.