Surds Quick Test

Simplify

\[ \sqrt{8} \]

\[ \begin{aligned} \sqrt{8} &= \sqrt{4\times 2} \\ &= \sqrt{4}\times\sqrt{2} \\ &= 2\sqrt{2} \end{aligned} \]

\[ \sqrt{200} \]

\[ \begin{aligned} \sqrt{200} &= \sqrt{100\times 2} \\ &= \sqrt{100}\times\sqrt{2} \\ &= 10\sqrt{2} \end{aligned} \]

\[ \sqrt{900} \]

\[ \begin{aligned} \sqrt{900} &= \sqrt{100\times 9} \\ &=\sqrt{100}\times\sqrt{9} \\ &=10\times 3 \\&= 30 \end{aligned} \]

\[ \sqrt{12} \]

\[ \begin{aligned} \sqrt{12} &= \sqrt{4\times 3} \\ &=\sqrt{4}\times\sqrt{3} \\ &=2\times\sqrt{3}\\&= 2\sqrt{3} \end{aligned} \]

\[ \sqrt{0.04} \]

\[ \sqrt{0.04} = \sqrt{0.01}\times \sqrt{4} = 0.1\times 2 = 0.2 \]

\[ \sqrt{27} \]

\[ \begin{aligned} \sqrt{27} &= \sqrt{9\times 3} \\ &=\sqrt{9}\times\sqrt{3} \\ &=3\times\sqrt{3}\\&= 3\sqrt{3} \end{aligned} \]

\[ \sqrt{50} \]

\[ \begin{aligned} \sqrt{50} &= \sqrt{25\times 2} \\ &=\sqrt{25}\times\sqrt{2} \\ &=5\times\sqrt{2}\\&= 5\sqrt{2} \end{aligned} \]

\[ \sqrt{63} \]

\[ \begin{aligned} \sqrt{63} &= \sqrt{9\times 7} \\ &=\sqrt{9}\times\sqrt{7} \\ &=3\times\sqrt{7}\\&= 3\sqrt{7} \end{aligned} \]

\[ 5\sqrt{8} \]

\[ \begin{aligned} 5\sqrt{8} &= 5\sqrt{4\times 2} \\ &=5\times\sqrt{4}\times\sqrt{2} \\ &=5\times2\times \sqrt{2}\\&= 10\times\sqrt{2}\\ &=10\sqrt{2}\end{aligned} \]

\[ 2\sqrt{98} \]

\[ \begin{aligned} 2\sqrt{98} &= 2\sqrt{49\times 2} \\ &=2\times\sqrt{49}\times\sqrt{2} \\ &=2\times 7\times \sqrt{2}\\&= 14\times\sqrt{2}\\ &=14\sqrt{2}\end{aligned} \]

Calculate

Express your answer in simplest form

\[ \sqrt{75} - \sqrt{12} \]

\[ \begin{aligned} \sqrt{75} - \sqrt{12} &= \sqrt{25\times 3} -\sqrt{4\times 3} \\ &=5\sqrt{3} - 2\sqrt{3} \\ &=3\sqrt{3}\end{aligned} \]

\[ \sqrt{18} + \sqrt{2} \]

\[ \begin{aligned} \sqrt{18} + \sqrt{2} &= \sqrt{9\times 2} +\sqrt{2} \\ &=3\sqrt{2} + \sqrt{2} \\ &=4\sqrt{2}\end{aligned} \]

\[ (\sqrt{2}-1)^2 \]

\[ \begin{aligned} (\sqrt{2}-1)^2 &= (\sqrt{2}-1)(\sqrt{2}-1) \\ &= \sqrt{2}\times\sqrt{2}-\sqrt{2}-\sqrt{2}+1 \\ &= 2-2\sqrt{2}+1 \\ &= 3-2\sqrt{2} \end{aligned} \]

\[ \frac{-4+\sqrt{8}}{2} \]

\[ \begin{aligned} \frac{-4+\sqrt{8}}{2} &= \frac{-4+2\sqrt{2}}{2} \\&= \frac{-2(2+\sqrt{2})}{2}\\ &= -2+\sqrt{2} \end{aligned} \]

Rationalise the denominator

\[ \frac{15}{\sqrt{2}} \]

\[ \begin{aligned} \frac{15}{\sqrt{2}} &= \frac{15}{\sqrt{2}}\times\frac{\sqrt{2}}{\sqrt{2}} \\ &= \frac{15\sqrt{2}}{\sqrt{2}\times\sqrt{2}} \\ &= \frac{15\sqrt{2}}{2} \end{aligned} \]

\[ \frac{1}{\sqrt{3}-\sqrt{2}} \]

\[ \begin{aligned} \frac{1}{\sqrt{3}-\sqrt{2}} &= \frac{1}{(\sqrt{3}-\sqrt{2})}\times\frac{(\sqrt{3}+\sqrt{2})}{(\sqrt{3}+\sqrt{2})} \\ &= \frac{\sqrt{3}+\sqrt{2}}{(\sqrt{3}-\sqrt{2})(\sqrt{3}+\sqrt{2})} \\ &= \frac{\sqrt{3}+\sqrt{2}}{\sqrt{3}\times\sqrt{3}-\sqrt{2}\times\sqrt{2}} \\ &= \frac{\sqrt{3}+\sqrt{2}}{3-2} \\ &= \frac{\sqrt{3}+\sqrt{2}}{1} \\ &= \sqrt{3}+\sqrt{2} \end{aligned} \]

\[ \frac{\sqrt{2}+1}{\sqrt{2}-1} \]

\[ \begin{aligned} \frac{\sqrt{2}+1}{\sqrt{2}-1} &= \frac{(\sqrt{2}+1)}{(\sqrt{2}-1)} \times \frac{(\sqrt{2}+1)}{(\sqrt{2}+1)} \\ &= \frac{(\sqrt{2}+1)(\sqrt{2}+1)}{2-1} \\ &= \frac{2+2\sqrt{2}+1}{1} \\ &= 3+2\sqrt{2} \end{aligned} \]

\[ \sqrt{\frac{9}{10}} \]

\[ \begin{aligned} \sqrt{\frac{9}{10}} &= \frac{\sqrt{9}}{\sqrt{10}} \\ &= \frac{3}{\sqrt{10}} \\ &= \frac{3}{\sqrt{10}} \times \frac{\sqrt{10}}{\sqrt{10}} \\ &=\frac{3\sqrt{10}}{10} \end{aligned} \]