Solve:$ \frac{6-4 \sqrt{5}+3 \sqrt{5}-10}{9-6 \sqrt{5}+6 \sqrt{5}-4 \sqrt{25}} $

Given:

\( \frac{6-4 \sqrt{5}+3 \sqrt{5}-10}{9-6 \sqrt{5}+6 \sqrt{5}-4 \sqrt{25}} \)

To do:

We have to evaluate \( \frac{6-4 \sqrt{5}+3 \sqrt{5}-10}{9-6 \sqrt{5}+6 \sqrt{5}-4 \sqrt{25}} \).

Solution:

We know that,

$(a-b)^2=a^2-2ab+b^2$

Therefore,

$\frac{6-4 \sqrt{5}+3 \sqrt{5}-10}{9-6 \sqrt{5}+6 \sqrt{5}-4 \sqrt{25}}$

$=\frac{6-\sqrt{5}-10}{9-4\times5}$

$=\frac{6-10-\sqrt{5}}{9-20}$

$=\frac{-4-\sqrt5}{-11}$

$=\frac{-(4+\sqrt5)}{-(11)}$

$=\frac{4+\sqrt{5}}{11}$