Question

Simplify $\frac{5-\sqrt{10}}{5+\sqrt{10}}$ - $\frac{5+\sqrt{10}}{5-\sqrt{10}}$

• A. $20\sqrt{10}$
• B. 35
• C. $\frac{4}{3}\sqrt{10}$
• D. -$\frac{4}{3}\sqrt{10}$

Answer 1

The correct option is D

-$\frac{4}{3}\sqrt{10}$

Given that
$\frac{5-\sqrt{10}}{5+\sqrt{10}}$ - $\frac{5+\sqrt{10}}{5-\sqrt{10}}$
Multiplying both numerator and denominator by the rationalising factor of the denominators , we have
= $\frac{5-\sqrt{10}}{5+\sqrt{10}}$ $\times$ $\frac{5-\sqrt{10}}{5-\sqrt{10}}$ - $\frac{5+=\sqrt{10}}{5-\sqrt{10}}$ $\times$ $\frac{5+\sqrt{10}}{5+\sqrt{10}}$

=$\frac{(5-\sqrt{10}{)}^2}{25-\sqrt{(}10{)}^2}$ - $\frac{(5+\sqrt{10}{)}^2}{25-\sqrt{(}10{)}^2}$

= $\frac{25-10\sqrt{10}+10}{15}$ - $\frac{25+10\sqrt{10}+10}{15}$

= $\frac{25-10\sqrt{10}+10-25-10\sqrt{10}-10}{15}$

= -$\frac{20\sqrt{10}}{15}$

= -$\frac{4}{3}\sqrt{10}$

$\frac{5-\sqrt{10}}{5+\sqrt{10}}$ - $\frac{5+\sqrt{10}}{5-\sqrt{10}}$ = = -$\frac{4}{3}\sqrt{10}$