Find the value of: $\frac{\sqrt{45} + \sqrt{18}} {\sqrt{7+2\sqrt{10}}}$

Question

Find the value of:$$\frac{\sqrt{45} + \sqrt{18}} {\sqrt{7+2\sqrt{10}}}$$

I need help solving this question. Every reply is appreciated.

Thanks!

Answer 1

We have:

$$\frac{\sqrt{45} + \sqrt{18}} {\sqrt{7+2\sqrt{10}}}$$

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Then we will square the fraction: $$ = \sqrt{\frac{(\sqrt{45}+\sqrt{18})^2} {(\sqrt{7+2\sqrt{10}})^2}}$$

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Finish the squares: $$=\sqrt{\frac{45+18\sqrt{10} + 18} {7+2\sqrt{10}}}$$

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Simplify: $$=\sqrt{\frac{63+18\sqrt{10}} {7+2\sqrt{10}}}$$

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Factor: $$\sqrt{\frac{9(7+2\sqrt{10})} {7+2\sqrt{10}}}$$

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Crossing Out: $$\sqrt{9}$$

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Which is equal to:

$$\sqrt{9} = 3$$

Answer 2

Hint:

$$\frac{\sqrt{45} + \sqrt{18}} {\sqrt{7+2\sqrt{10}}} = \frac{3 \sqrt{5}+3\sqrt{2}}{\sqrt{\left(\sqrt{2}\right)^2+\left(\sqrt{5}\right)^2+2\,\sqrt{2}\sqrt{5}}}$$