<s>[INST] How to simplify an expression involving several square roots without a calculator? $$\frac{5 \sqrt{7}}{4\sqrt{3\sqrt{5}}-4\sqrt{2\sqrt{5}}}- \frac{4 \sqrt{5}}{\sqrt{3\sqrt{5}}-\sqrt{2\sqrt{5}}}$$
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.
 [/INST] 
Let $\sqrt{a\sqrt{b}} = \sqrt{a}\sqrt{\sqrt{b}} = \sqrt{a}\cdot\sqrt[4]{b}$

$$\tag1
\frac{5 \sqrt{7}}{4\sqrt{3\sqrt{5}}-4\sqrt{2\sqrt{5}}}- \frac{4 \sqrt{5}}{\sqrt{3\sqrt{5}}-\sqrt{2\sqrt{5}}}$$
$$\tag2
\frac{5\sqrt{7}}{4\sqrt{3}\sqrt[4]{5}-4\sqrt{2}\sqrt[4]{5}}- \frac{4\sqrt{5}}{\sqrt{3}\sqrt[4]{5}-\sqrt{2}\sqrt[4]{5}}$$
$$\tag3
\left(\frac{5\sqrt{7}}{4(\sqrt{3}-\sqrt{2})}- \frac{4\sqrt{5}}{\sqrt{3}-\sqrt{2}}\right)\frac{1}{\sqrt[4]{5}}$$
$$\tag4
\left(\frac{5 \sqrt{7} - 16 \sqrt{5}}{4(\sqrt{3}-\sqrt{2})}\right)\frac{1}{\sqrt[4]{5}}$$

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