Method 1
Find a² (by multiplying √8 + 2 by itself)

$\begin{array}{rcl} a^{2} & = & (\sqrt{8} + 2) (\sqrt{8} + 2) \\ = & \sqrt{8} \times \sqrt{8} + \sqrt{8} \times 2 + 2 \times \sqrt{8} + 2 \times 2 \end{array}$

[1]

Simplify the terms (using the surd law $\sqrt{a} \times \sqrt{a} = {(\sqrt{a})}^{2} = a$ for the first term)

$\begin{array}{rcl} a^{2} & = & 8 + 2 \sqrt{8} + 2 \sqrt{8} + 4 \\ = & 12 + 4 \sqrt{8} \end{array}$

Find b² (by multiplying √8 - 2 by itself) and do the same steps as above

$\begin{array}{rcl} b^{2} & = & (\sqrt{8} - 2) (\sqrt{8} - 2) \\ = & \sqrt{8} \times \sqrt{8} - \sqrt{8} \times 2 - 2 \times \sqrt{8} + 2 \times 2 \\ = & 8 - 2 \sqrt{8} - 2 \sqrt{8} + 4 \\ = & 12 - 4 \sqrt{8} \end{array}$

[1]

Find a² - b² by subtracting the two results above (using brackets around the results)
Collect like terms

$\begin{array}{rcl} a^{2} - b^{2} & = & (12 + 4 \sqrt{8}) - (12 - 4 \sqrt{8}) \\ = & 12 + 4 \sqrt{8} - 12 + 4 \sqrt{8} \\ = & 8 \sqrt{8} \end{array}$

[1]

The result must be given in the form c√2 where c is an integer (whole number)
Write $\sqrt{8}$ as $2 \sqrt{2}$ (from $\sqrt{8} = \sqrt{4 \times 2} = \sqrt{4} \sqrt{2} = 2 \sqrt{2}$ )

$\begin{array}{rcl} a^{2} - b^{2} & = & 8 \times 2 \sqrt{2} \\ = & 16 \sqrt{2} \end{array}$

$16 \sqrt{2}$ (where c = 16) [1]

Method 2
Use the "difference of two squares" to factorise a² - b²

$a^{2} - b^{2} = (a + b) (a - b)$

[1]

Substitute a = √8 + 2 and b = √8 - 2 into the expression and collect like terms

$\begin{array}{rcl} a^{2} - b^{2} & = & (\sqrt{8} + 2 + \sqrt{8} - 2) (\sqrt{8} + 2 - (\sqrt{8} - 2)) \\ = & (2 \sqrt{8}) (\sqrt{8} + 2 - \sqrt{8} + 2) \\ = & (2 \sqrt{8}) (4) \\ = & 8 \sqrt{8} \end{array}$

[1]

The result must be given in the form c√2 where c is an integer (whole number)
Write $\sqrt{8}$ as $2 \sqrt{2}$ (from $\sqrt{8} = \sqrt{4 \times 2} = \sqrt{4} \sqrt{2} = 2 \sqrt{2}$ )

$\begin{array}{rcl} a^{2} - b^{2} & = & 8 \times 2 \sqrt{2} \end{array}$

[1]

Simplify the final answer (using 8 × 2 = 16)

$16 \sqrt{2}$ (where c = 16) [1]

Surds (Cambridge O Level Maths)

Exam Questions