Question

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

Answer 1

A couple of ideas...

Explanation:

Given:

`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)`

`sqrt(a)/(sqrt(a)-sqrt(n)) = 1/(1-sqrt(n)/sqrt(a)) = 1/(1-sqrt(n/a))`

Option 1b - Multiply both numerator and denominator by `sqrt(a)`

`sqrt(a)/(sqrt(a)-sqrt(n)) = a/(a-sqrt(a)sqrt(n)) = a/(a-sqrt(an))`

Option 2 - Rationalise the denominator

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

`sqrt(a)/(sqrt(a)-sqrt(n)) = (sqrt(a)(sqrt(a)+sqrt(n)))/((sqrt(a)-sqrt(n))(sqrt(a)+sqrt(n)))`

`color(white)(sqrt(a)/(sqrt(a)-sqrt(n))) = (a+sqrt(a)sqrt(n))/(a-n)`

`color(white)(sqrt(a)/(sqrt(a)-sqrt(n))) = (a+sqrt(an))/(a-n)`

Answer 2

`(a + sqrt(an))/(a - n)`

Explanation:

That's conjugate surd..

See Process Below;

`sqrta/(sqrta - sqrtn)`

`sqrta/(sqrta - sqrtn) xx color(blue)((sqrta + sqrtn)/(sqrta + sqrtn)) -> "Conjugate"`

`(sqrta (sqrta + sqrtn))/((sqrta - sqrtn) (sqrta + sqrtn))`

`(a+ (sqrt(a xx n)))/(a - n)`

`(a + sqrt(an))/(a - n) -> "Simplified"`