Powers and Roots – Simplify an expression with roots – Exercise 5676

Exercise

Simplify the expression:

$\frac{\sqrt{5}}{2\sqrt{3}-1}-\frac{\sqrt{3}}{2\sqrt{5}+3}$

Final Answer

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Solution

Using Powers and Roots rules we get:

$\frac{\sqrt{5}}{2\sqrt{3}-1}-\frac{\sqrt{3}}{2\sqrt{5}+3}=$

$=\frac{\sqrt{5}(2\sqrt{3}+1)}{(2\sqrt{3}-1)(2\sqrt{3}+1)}-\frac{\sqrt{3}(2\sqrt{5}-3)}{(2\sqrt{5}+3)(2\sqrt{5}-3)}=$

$=\frac{\sqrt{5}(2\sqrt{3}+1)}{12-1}-\frac{\sqrt{3}(2\sqrt{5}-3)}{20-9}=$

$=\frac{1}{11}\cdot\sqrt{5}(2\sqrt{3}+1)-\frac{1}{11}\cdot\sqrt{3}(2\sqrt{5}-3)=$

$=\frac{1}{11}(2\sqrt{15}+\sqrt{5})-\frac{1}{11}(2\sqrt{15}-3\sqrt{3})=$

$=\frac{1}{11}(2\sqrt{15}+\sqrt{5}-2\sqrt{15}+3\sqrt{3})=$

$=\frac{1}{11}(\sqrt{5}+3\sqrt{3})$