Simplifying a square root fraction

$\begingroup$

Simplify the following

$$\frac{\sqrt{3}}{\sqrt{2}(\sqrt{6} - \sqrt{3})}$$

Apparently the answer is $\frac{1}{2} (2 + \sqrt{2})$ but can't for the life of me see how to get it. Any help is massively appreciated.

Thanks

$\endgroup$ 1

2 Answers

$\begingroup$

$\dfrac{\sqrt 3}{\sqrt 2 (\sqrt 6 - \sqrt 3)} = \dfrac{\sqrt 3}{\sqrt 2 \sqrt 3(\sqrt2 - 1)} = \dfrac{1}{\sqrt 2 (\sqrt 2 - 1)} = \dfrac{1}{(2 - \sqrt 2)} = \dfrac{(2 + \sqrt 2)}{(2 - \sqrt 2)(2 + \sqrt 2)} = \dfrac{2 + \sqrt 2}{2}$

The reason we write it this way is that it is bad practice to have roots at the denominator. To get rid of $a+b \sqrt c$ at the denominator, simply multiply the numerator and the denominator by $a-b \sqrt c$.

$\endgroup$ 2 $\begingroup$

$\frac{\sqrt{3}}{\sqrt{2}(\sqrt{6}-\sqrt{3})}=\frac{\sqrt{1}}{\sqrt{2}(\sqrt{2}-\sqrt{1})}=\frac{\sqrt{2}+\sqrt{1}}{\sqrt{2}}=\frac12(2+\sqrt{2}).$

$\endgroup$ 1

Your Answer

Sign up or log in

Sign up using Google Sign up using Facebook Sign up using Email and Password Submit

Post as a guest

Post Your Answer Discard

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy