Simplifying surds

Surds can be simplified if the number in the surd has a square number as a factor.

Remember these general rules:

\sqrt{a} \times \sqrt{a} = a

\sqrt{ab} = \sqrt{a} \times \sqrt{b}

\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}

Examples

Simplify \sqrt{12}.

4 is a factor of 12 so we can write \sqrt{12} = \sqrt{(4\times3)} = \sqrt{4}\times\sqrt{3}

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

Simplify \sqrt{10} \times \sqrt{5}

\sqrt{10} \times \sqrt{5} = \sqrt{50}

50 = 25 \times 2, so we can write \sqrt{50} = \sqrt{25 \times 2} = \sqrt{25} \times \sqrt{2} = 5\sqrt{2}

Simplify \frac{\sqrt{12}}{\sqrt{6}}

\frac{\sqrt{12}}{\sqrt{6}} = \sqrt{\frac{12}{6}} = \sqrt{2}

Question

Simplify the following surds:

  1. \sqrt{8}
  2. \sqrt{8} \times \sqrt{4}
  3. \sqrt{18}
  4. \frac{\sqrt{18}}{\sqrt{9}}
  1. \sqrt{4} \times \sqrt{2} = 2 \sqrt{2}
  2. \sqrt{(8 \times 4)} = \sqrt{32} = \sqrt{(2 \times 16)} = 4 \sqrt{2}
  3. \sqrt{2} \times \sqrt{9} = 3 \sqrt{2}
  4. \frac{\sqrt{18}}{{9}} = \sqrt{2}
curriculum-key-fact
Answers left in surd form are exact answers.