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

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

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Answer 1 · 2017-10-25T20:32:14

A couple of ideas...

Explanation:

Given:

$\frac{\sqrt{a}}{\sqrt{a} - \sqrt{n}}$ $sqrt(a)/(sqrt(a)-sqrt(n))$

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 $\sqrt{a}$ $sqrt(a)$

$\frac{\sqrt{a}}{\sqrt{a} - \sqrt{n}} = \frac{1}{1 - \frac{\sqrt{n}}{\sqrt{a}}} = \frac{1}{1 - \sqrt{\frac{n}{a}}}$ $sqrt(a)/(sqrt(a)-sqrt(n)) = 1/(1-sqrt(n)/sqrt(a)) = 1/(1-sqrt(n/a))$

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

$\frac{\sqrt{a}}{\sqrt{a} - \sqrt{n}} = \frac{a}{a - \sqrt{a} \sqrt{n}} = \frac{a}{a - \sqrt{a n}}$ $sqrt(a)/(sqrt(a)-sqrt(n)) = a/(a-sqrt(a)sqrt(n)) = a/(a-sqrt(an))$

Option 2 - Rationalise the denominator

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

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

$\frac{\sqrt{a}}{\sqrt{a} - \sqrt{n}} = \frac{a + \sqrt{a} \sqrt{n}}{a - n}$ $color(white)(sqrt(a)/(sqrt(a)-sqrt(n))) = (a+sqrt(a)sqrt(n))/(a-n)$

$\frac{\sqrt{a}}{\sqrt{a} - \sqrt{n}} = \frac{a + \sqrt{a n}}{a - n}$ $color(white)(sqrt(a)/(sqrt(a)-sqrt(n))) = (a+sqrt(an))/(a-n)$

Answer 2 · 2017-10-25T20:41:16

$\frac{a + \sqrt{a n}}{a - n}$ $(a + sqrt(an))/(a - n)$

Explanation:

That's conjugate surd..

See Process Below;

$\frac{\sqrt{a}}{\sqrt{a} - \sqrt{n}}$ $sqrta/(sqrta - sqrtn)$

$\frac{\sqrt{a}}{\sqrt{a} - \sqrt{n}} \times \frac{\sqrt{a} + \sqrt{n}}{\sqrt{a} + \sqrt{n}} \to Conjugate$ $sqrta/(sqrta - sqrtn) xx color(blue)((sqrta + sqrtn)/(sqrta + sqrtn)) -> "Conjugate"$

$\frac{\sqrt{a} (\sqrt{a} + \sqrt{n})}{(\sqrt{a} - \sqrt{n}) (\sqrt{a} + \sqrt{n})}$ $(sqrta (sqrta + sqrtn))/((sqrta - sqrtn) (sqrta + sqrtn))$

$\frac{a + (\sqrt{a \times n})}{a - n}$ $(a+ (sqrt(a xx n)))/(a - n)$

$\frac{a + \sqrt{a n}}{a - n} \to Simplified$ $(a + sqrt(an))/(a - n) -> "Simplified"$

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

2 Answers

Explanation:

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