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

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!

We have:

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

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

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

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

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

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

Which is equal to:

$$\sqrt{9} = 3$$

Hint:

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

• Wish the downvoter had left a comment why.
– dxiv
Feb 4, 2017 at 4:31
• Nice Hint. (+1). And who are these nefarious down voters? Feb 4, 2017 at 5:36
• @dxiv \ Don't worry about it, man. There are people in this world who pride themselves on being walking, talking toothaches. Feb 4, 2017 at 7:29