How do you simplify (sqrt5+sqrt3)/(sqrt3-sqrt5?

Mar 21, 2017

$- \left(4 + \sqrt{15}\right)$

Explanation:

Expression =(sqrt5+sqrt3)/(sqrt3-sqrt5

To simplify the expression we first need to rationalise the denominator.

Remember: $\left(a + b\right) \left(a - b\right) = {a}^{2} - {b}^{2}$

Hence: $\left(\sqrt{a} + \sqrt{b}\right) \left(\sqrt{a} - \sqrt{b}\right) = a - b$

Multiply our expression by: $\frac{\sqrt{3} + \sqrt{5}}{\sqrt{3} + \sqrt{5}} = 1$

$\therefore$ Expression $= \frac{\left(\sqrt{5} + \sqrt{3}\right) \left(\sqrt{3} + \sqrt{5}\right)}{\left(\sqrt{3} - \sqrt{5}\right) \left(\sqrt{3} + \sqrt{5}\right)}$

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

$= \frac{\sqrt{15} + 5 + 3 + \sqrt{15}}{-} 2$

$= \frac{8 + 2 \sqrt{15}}{-} 2$

$= - \left(4 + \sqrt{15}\right)$