Find the value of:$$\frac{\sqrt{45} + \sqrt{18}} {\sqrt{7+2\sqrt{10}}}$$
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Find the value of: $\frac{\sqrt{45} + \sqrt{18}} {\sqrt{7+2\sqrt{10}}}$
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$\begingroup$
contest-math
square-numbers
sums-of-squares
2 Answers
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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:
$$\sqrt{9} = 3$$
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0Very well laid out. (+1) – 2017-02-04
5
Hint:
$$\frac{\sqrt{45} + \sqrt{18}} {\sqrt{7+2\sqrt{10}}} = \frac{3 \sqrt{5}+3\sqrt{2}}{\sqrt{\left(\sqrt{2}\right)^2+\left(\sqrt{5}\right)^2+2\,\sqrt{2}\sqrt{5}}}$$
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0Wish the downvoter had left a comment why. – 2017-02-04
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1Nice Hint. (+1). And who are these nefarious down voters? – 2017-02-04
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1@dxiv \ Don't worry about it, man. There are people in this world who pride themselves on being walking, talking toothaches. – 2017-02-04