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Surds

Secondary 3 Additional Math practice

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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.

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What this topic covers

Worked examples

Example 1

Simplify 6×40\sqrt{6} \times \sqrt{40}.
Show the worked solution
  • 6×40=6×40product rule for surds\sqrt{6} \times \sqrt{40} = \sqrt{6 \times 40} \quad \scriptsize\textit{product rule for surds}
  • =240= \sqrt{240}
  • =16×15= \sqrt{16 \times 15}
  • =415= 4\sqrt{15}

Answer: 4√154√15

Example 2

Simplify (5−2)(4+22)(5 - \sqrt{2})(4 + 2\sqrt{2}). Express your answer in the form a+b2a + b\sqrt{2}.
Show the worked solution
  • (5−2)(4+22)expand using FOIL(5 - \sqrt{2})(4 + 2\sqrt{2}) \quad \scriptsize\textit{expand using FOIL}
  • 5×4=205 \times 4 = 20
  • 5×22=1025 \times 2\sqrt{2} = 10\sqrt{2}
  • −2×4=−42-\sqrt{2} \times 4 = -4\sqrt{2}
  • −2×22=−2×2-\sqrt{2} \times 2\sqrt{2} = -2 \times 2
  • −2×2=−4-2 \times 2 = -4
  • (5−2)(4+22)=20−4+(10−4)2combine like terms(5 - \sqrt{2})(4 + 2\sqrt{2}) = 20 - 4 + (10 - 4)\sqrt{2} \quad \scriptsize\textit{combine like terms}
  • =16+62= 16 + 6\sqrt{2}

Answer: "rational":"16","irrational":"6"{"rational":"16","irrational":"6"}

Example 3

Rationalise the denominator of 22+3\frac{2}{2 + \sqrt{3}} and express in the form a+b3a + b\sqrt{3}.
Show the worked solution
  • 22+3×2−32−3multiply by conjugate\frac{2}{2 + \sqrt{3}} \times \frac{2 - \sqrt{3}}{2 - \sqrt{3}} \quad \scriptsize\textit{multiply by conjugate}
  • =2(2−3)(2+3)(2−3)= \frac{2(2 - \sqrt{3})}{(2 + \sqrt{3})(2 - \sqrt{3})}
  • =4−2322−(3)2difference of squares= \frac{4 - 2\sqrt{3}}{2^2 - (\sqrt{3})^2} \quad \scriptsize\textit{difference of squares}
  • =4−234−3= \frac{4 - 2\sqrt{3}}{4 - 3}
  • =4−23= 4 - 2\sqrt{3}

Answer: "rational":"4","irrational":"−2"{"rational":"4","irrational":"-2"}

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.

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