# How do you simplify \frac { \sqrt { a } } { \sqrt { a } - \sqrt { n } }?

Oct 25, 2017

A couple of ideas...

#### Explanation:

Given:

$\frac{\sqrt{a}}{\sqrt{a} - \sqrt{n}}$

I am not sure what we would consider a simplified expression, but here are some things we can do:

Option 1a - Divide numerator and denominator by $\sqrt{a}$

$\frac{\sqrt{a}}{\sqrt{a} - \sqrt{n}} = \frac{1}{1 - \frac{\sqrt{n}}{\sqrt{a}}} = \frac{1}{1 - \sqrt{\frac{n}{a}}}$

Option 1b - Multiply both numerator and denominator by $\sqrt{a}$

$\frac{\sqrt{a}}{\sqrt{a} - \sqrt{n}} = \frac{a}{a - \sqrt{a} \sqrt{n}} = \frac{a}{a - \sqrt{a n}}$

Option 2 - Rationalise the denominator

We can rationalise the denominator by multiplying both numerator and denominator by the radical conjugate of the denominator...

$\frac{\sqrt{a}}{\sqrt{a} - \sqrt{n}} = \frac{\sqrt{a} \left(\sqrt{a} + \sqrt{n}\right)}{\left(\sqrt{a} - \sqrt{n}\right) \left(\sqrt{a} + \sqrt{n}\right)}$

$\textcolor{w h i t e}{\frac{\sqrt{a}}{\sqrt{a} - \sqrt{n}}} = \frac{a + \sqrt{a} \sqrt{n}}{a - n}$

$\textcolor{w h i t e}{\frac{\sqrt{a}}{\sqrt{a} - \sqrt{n}}} = \frac{a + \sqrt{a n}}{a - n}$

Oct 25, 2017

$\frac{a + \sqrt{a n}}{a - n}$

#### Explanation:

That's conjugate surd..

See Process Below;

$\frac{\sqrt{a}}{\sqrt{a} - \sqrt{n}}$

$\frac{\sqrt{a}}{\sqrt{a} - \sqrt{n}} \times \textcolor{b l u e}{\frac{\sqrt{a} + \sqrt{n}}{\sqrt{a} + \sqrt{n}}} \to \text{Conjugate}$

$\frac{\sqrt{a} \left(\sqrt{a} + \sqrt{n}\right)}{\left(\sqrt{a} - \sqrt{n}\right) \left(\sqrt{a} + \sqrt{n}\right)}$

$\frac{a + \left(\sqrt{a \times n}\right)}{a - n}$

$\frac{a + \sqrt{a n}}{a - n} \to \text{Simplified}$