Question

How do you simplify $\dfrac{{16}}{{\sqrt 5 - \sqrt 7 }}$

Answer 1

Hint: In order to simplify this question we will first remove the term that are present in the denominator, for that we will multiply and divide by the conjugate of the numerator and then the denominator will be converted to any integer and hence it will be simplified.

Complete step-by-step solution:
For solving these type of question we should be strike that the denominator’s root is the main typical term which has to be in integer form for this we will be multiplying and dividing the conjugate of the denominator so the conjugate of the denominator will be:
$\Rightarrow \sqrt 5 + \sqrt 7 $
We will be multiplying it in the above term:
$\Rightarrow \dfrac{{16\left( {\sqrt 5 + \sqrt 7 } \right)}}{{\sqrt 5 - \sqrt 7 \left( {\sqrt 5 + \sqrt 7 } \right)}}$
Now as we can see that in denominator the there is a identity which can be applied, the identity is $\Rightarrow {a^2} - {b^2} = (a + b)(a - b)$
Now applying this to the main term’s denominator:
$\Rightarrow \dfrac{{16(\sqrt 5 + \sqrt 7 )}}{{{5^2} - {7^2}}}$
Now on further solving the denominator we will be getting:
$\Rightarrow \dfrac{{16(\sqrt 5 + \sqrt {7)} }}{{25 - 47}}$
On further solving:
$\Rightarrow \dfrac{{16(\sqrt 5 + \sqrt 7 )}}{{ - 24}}$
Now we will be dividing the numerator and denominator by 8 we will be getting:
$\Rightarrow \dfrac{{ - 2(\sqrt 5 + \sqrt 7 )}}{3}$

Hence the correct answer is $\dfrac{{ - 2(\sqrt 5 + \sqrt 7 )}}{3}$

Note: While solving these types of questions we should always remember that we will always multiply the terms by the numerator and denominator to make the denominator rootless will be the conjugate of the denominator’s term if there is + sign then we will using – sign and vice versa.