Question
Evaluate $\frac{15}{\sqrt{10}+\sqrt{20}+\sqrt{40}-\sqrt{5}-\sqrt{80}},$ it being given that $\sqrt{5}=2.236$ and $\sqrt{10}=3.162$
✓
Answer
$\frac{15}{\sqrt{10}+\sqrt{20}+\sqrt{40}-\sqrt{5}-\sqrt{80}}$
$=\frac{15}{\sqrt{10}+\sqrt{4\times5}+\sqrt{4\times10}-\sqrt{5}-\sqrt{16\times5}}$
$=\frac{15}{\sqrt{10}+2\sqrt{5}+2\sqrt{10}-\sqrt{5}-4\sqrt{5}}$
$=\frac{15}{3\sqrt{10}-3\sqrt{5}}$
$=\frac{5}{\sqrt{10}-\sqrt{5}}$
$=\frac{5}{\sqrt{10}-\sqrt{5}}\times\frac{\sqrt{10}+\sqrt{5}}{\sqrt{10}+\sqrt{5}}$
$=\frac{5\big(\sqrt{10}+\sqrt{5}\big)}{\big(\sqrt{10}\big)^2-\big(\sqrt{5}\big)^2}$
$=\frac{5\big(\sqrt{10}+\sqrt{5}\big)}{10-5}$
$=\frac{5\big(\sqrt{10}+\sqrt{5}\big)}{5}$
$=\sqrt{10}+\sqrt{5}$
$=3.162+2.236$ $=5.398$
$=\frac{15}{\sqrt{10}+\sqrt{4\times5}+\sqrt{4\times10}-\sqrt{5}-\sqrt{16\times5}}$
$=\frac{15}{\sqrt{10}+2\sqrt{5}+2\sqrt{10}-\sqrt{5}-4\sqrt{5}}$
$=\frac{15}{3\sqrt{10}-3\sqrt{5}}$
$=\frac{5}{\sqrt{10}-\sqrt{5}}$
$=\frac{5}{\sqrt{10}-\sqrt{5}}\times\frac{\sqrt{10}+\sqrt{5}}{\sqrt{10}+\sqrt{5}}$
$=\frac{5\big(\sqrt{10}+\sqrt{5}\big)}{\big(\sqrt{10}\big)^2-\big(\sqrt{5}\big)^2}$
$=\frac{5\big(\sqrt{10}+\sqrt{5}\big)}{10-5}$
$=\frac{5\big(\sqrt{10}+\sqrt{5}\big)}{5}$
$=\sqrt{10}+\sqrt{5}$
$=3.162+2.236$ $=5.398$
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