Surds practice questions

Secondary 3 topics from the Additional Mathematics 4049 syllabus. Every question below has a full worked solution.

When a student gets a question wrong in Math Amigo, it does not show the answer straight away. It reads the working, identifies the step that went wrong, and asks a question that points at the correction. The worked solution is there once a genuine attempt has been made.

Practise surds free

What this topic covers

Worked examples

Example 1

Simplify 483\sqrt{\frac{48}{3}}.
Show the worked solution
  • $\sqrt{\frac{48}{3}} = \frac{\sqrt{48}}{\sqrt{3}} \quad \scriptsize\textit{quotient rule for surds}$
  • $= \sqrt{16}$
  • $= 4$

Answer: 44

Example 2

Simplify (3−7)(1−7)(3 - \sqrt{7})(1 - \sqrt{7}). Express your answer in the form a+b7a + b\sqrt{7}.
Show the worked solution
  • $(3 - \sqrt{7})(1 - \sqrt{7}) \quad \scriptsize\textit{expand using FOIL}$
  • $3 \times 1 = 3$
  • $3 \times (-1)\sqrt{7} = -3\sqrt{7}$
  • $(-1)\sqrt{7} \times 1 = -\sqrt{7}$
  • $(-1)\sqrt{7} \times (-1)\sqrt{7} = 1 \times 7$
  • $1 \times 7 = 7$
  • $= 3 + 7 + (-3 - 1)\sqrt{7} \quad \scriptsize\textit{combine like terms}$
  • $= 10 - 4\sqrt{7}$

Answer: "rational":"10","irrational":"−4"{"rational":"10","irrational":"-4"}

Example 3

Rationalise the denominator of 32−7\frac{3}{2 - \sqrt{7}} and express in the form a+b7a + b\sqrt{7}.
Show the worked solution
  • $\frac{3}{2 - \sqrt{7}} \times \frac{2 + \sqrt{7}}{2 + \sqrt{7}} \quad \scriptsize\textit{multiply by conjugate}$
  • $= \frac{3(2 + \sqrt{7})}{(2 - \sqrt{7})(2 + \sqrt{7})}$
  • $= \frac{6 + 3\sqrt{7}}{2^2 - (-\sqrt{7})^2} \quad \scriptsize\textit{difference of squares}$
  • $= \frac{6 + 3\sqrt{7}}{4 - 7}$
  • $= \frac{6 + 3\sqrt{7}}{-3}$
  • $= -2 - \sqrt{7}$

Answer: "rational":"−2","irrational":"−1"{"rational":"-2","irrational":"-1"}

Questions parents ask

Is it really free?

The free plan is permanent and does not ask for a card. It has daily limits on the AI feedback, and the paid plans lift them.

Which syllabus does this follow?

Additional Mathematics (4049), for Singapore secondary students. Math Amigo covers Secondary 1 to 4, both O-Level Mathematics and Additional Mathematics.

Will it just show my child the answer?

No. On a wrong answer it looks at the working, points to the step that went wrong, and asks a question that helps the student find the correction. The full worked solution is available after a genuine attempt.

Other Secondary 3 Additional Math topics