# What is [5(square root of 5) + 3(square root of 7)] / [ 4(square root of 7) - 3 (square root of 5) ]?

Sep 15, 2015

$\frac{159 + 29 \sqrt{35}}{47}$
$\textcolor{w h i t e}{\text{XXXXXXXX}}$assuming I haven't made any arithmetic errors

#### Explanation:

(5(sqrt(5))+3(sqrt(7)))/(4(sqrt(7))-3(sqrt(5))

Rationalize the denominator by multiplying by the conjugate:

$= \frac{5 \left(\sqrt{5}\right) + 3 \left(\sqrt{7}\right)}{4 \left(\sqrt{7}\right) - 3 \left(\sqrt{5}\right)} \times \frac{4 \left(\sqrt{7}\right) + 3 \left(\sqrt{5}\right)}{4 \left(\sqrt{7}\right) + 3 \left(\sqrt{5}\right)}$

$= \frac{20 \sqrt{35} + 15 \left({\left(\sqrt{5}\right)}^{2}\right) + 12 \left({\left(\sqrt{7}\right)}^{2}\right) + 9 \sqrt{35}}{16 \left({\left(\sqrt{7}\right)}^{2}\right) - 9 \left({\left(\sqrt{5}\right)}^{2}\right)}$

$= \frac{29 \sqrt{35} + 15 \left(5\right) + 12 \left(7\right)}{16 \left(7\right) - 9 \left(5\right)}$

$= \frac{29 \sqrt{35} + 75 + 84}{112 - 45}$

$= \frac{159 + 29 \sqrt{35}}{47}$