How to simplify an expression involving several square roots without a calculator?

Question

$$\frac{5 \sqrt{7}}{4\sqrt{3\sqrt{5}}-4\sqrt{2\sqrt{5}}}- \frac{4 \sqrt{5}}{\sqrt{3\sqrt{5}}-\sqrt{2\sqrt{5}}}$$ This type of questions are common in the university entrance examinations in our country but the calculators are not allowed can someone help me to find the way to simplify the expression.

Answer 1

Assuming that your expression contains a typo :)

Actually we have to simplify $$\frac{5 \sqrt{5}}{4\left(\sqrt{3\sqrt{5}}-\sqrt{2\sqrt{5}}\right)}- \frac{4 \sqrt{5}}{\sqrt{3\sqrt{5}}-\sqrt{2\sqrt{5}}}$$

Notice that both fractions can be simplified multiplying numerator and denominator by

$\left(\sqrt{3\sqrt{5}}+\sqrt{2\sqrt{5}}\right)$

Indeed $$\left(\sqrt{3\sqrt{5}}-\sqrt{2\sqrt{5}}\right)\left(\sqrt{3\sqrt{5}}+\sqrt{2\sqrt{5}}\right)=3\sqrt{5}-2\sqrt{5}=\sqrt{5}$$ the expression becomes $$\frac{\left(5 \sqrt{5}\right)\left(\sqrt{3\sqrt{5}}+\sqrt{2\sqrt{5}}\right)}{4\sqrt 5}- \frac{4 \sqrt{5}\left(\sqrt{3\sqrt{5}}-\sqrt{2\sqrt{5}}\right)}{\sqrt 5}=\\=\frac{5 \left(\sqrt{3\sqrt{5}}+\sqrt{2\sqrt{5}}\right)}{4}- 4\left(\sqrt{3\sqrt{5}}-\sqrt{2\sqrt{5}}\right)=\frac{21 \sqrt{2 \sqrt{5}}}{4}-\frac{11 \sqrt{3 \sqrt{5}}}{4}$$

If the $\sqrt{7}$ is not a typo the numerator is a bit more complicated, but the basics are always the same

Answer 2

Hint: apply $\frac{1}{\sqrt{a} - \sqrt{b}}=\frac{\sqrt{a} + \sqrt{b}}{a - b}$.

Answer 3

$$\frac{5 \sqrt{7}}{4\sqrt{3\sqrt{5}}-4\sqrt{2\sqrt{5}}}- \frac{4 \sqrt{5}}{\sqrt{3\sqrt{5}}-\sqrt{2\sqrt{5}}}=\frac{5\sqrt7-16\sqrt5}{4(\sqrt{3\sqrt5}-\sqrt{2\sqrt5})}.$$

Answer 4

Let $\sqrt{a\sqrt{b}} = \sqrt{a}\sqrt{\sqrt{b}} = \sqrt{a}\cdot\sqrt[4]{b}$

$$\tag1 \frac{5 \sqrt{7}}{4\sqrt{3\sqrt{5}}-4\sqrt{2\sqrt{5}}}- \frac{4 \sqrt{5}}{\sqrt{3\sqrt{5}}-\sqrt{2\sqrt{5}}}$$

$$\tag2 \frac{5\sqrt{7}}{4\sqrt{3}\sqrt[4]{5}-4\sqrt{2}\sqrt[4]{5}}- \frac{4\sqrt{5}}{\sqrt{3}\sqrt[4]{5}-\sqrt{2}\sqrt[4]{5}}$$

$$\tag3 \left(\frac{5\sqrt{7}}{4(\sqrt{3}-\sqrt{2})}- \frac{4\sqrt{5}}{\sqrt{3}-\sqrt{2}}\right)\frac{1}{\sqrt[4]{5}}$$

$$\tag4 \left(\frac{5 \sqrt{7} - 16 \sqrt{5}}{4(\sqrt{3}-\sqrt{2})}\right)\frac{1}{\sqrt[4]{5}}$$