Question

Find the fourth root of $124 - 32\sqrt {15} $
A. $\sqrt 5 + \sqrt 3 $
B. $\sqrt 5 - \sqrt 3 $
C. $\sqrt {31} - \sqrt 4 $
D. $\sqrt {60} - \sqrt {64} $

Answer 1

Hint: In this question we have to convert the given number $124 - 32\sqrt {15} $ into a perfect square identity for solving it further. Here the most important thing is the perfect square of numbers. We have to find the fourth root so, don’t forget to multiply the power with $\dfrac{1}{4}$ at the end.

Complete step-by-step answer:
We have $124 - 32\sqrt {15} $
We can write this number as
$60 + 64 - 32\sqrt {15} $
When factoring it further we get
${(2\sqrt {15} )^2} + {8^2} - 8.2.2\sqrt {15} $
Here we can clearly see that the above equation is making the identity of
${(a - b)^2} = {a^2} - 2ab + {b^2}$
$a = 2\sqrt {15} ,b = 8,2ab = 2.8.2\sqrt {15} $
So, we have ${(2\sqrt {15} - 8)^2}$
We have to find the fourth root and now we have the root of the equation
We can write ${(2\sqrt {15} - 8)^2}$it as
${(2.\sqrt 3 \sqrt 5 - {(\sqrt 3 )^2} + {(\sqrt 5 )^2})^2}$
Now we have again ${(a - b)^2} = {a^2} - 2ab + {b^2}$
 ${(2.\sqrt 3 \sqrt 5 - {(\sqrt 3 )^2} + {(\sqrt 5 )^2})^2}$ = ${({(\sqrt 5 - \sqrt 3 )^2})^2}$
After multiplying the power, we get
= ${(\sqrt 5 - \sqrt 3 )^4}$
For finding the fourth root we must multiply the power by $\dfrac{1}{4}$
= ${(\sqrt 5 - \sqrt 3 )^{4.\dfrac{1}{4}}}$
After cancelling the denominator and numerator in the power we get
= ${(\sqrt 5 - \sqrt 3 )^1}$
We can write the above numbers as
= $\sqrt 5 - \sqrt 3 $
 Hence, the fourth root of the $124 - 32\sqrt {15} $ is $\sqrt 5 - \sqrt 3 $

So, option B is the correct option.

Note: Try to make the numbers perfect square and use the identities to solve it further. Here students get confused between finding the values of a and b. Always remember some basic roots and squares. Whenever you have to find the fourth root, don’t forget to multiply the power with $\dfrac{1}{4}$. Do the calculation correctly and use the identity according to the equation given in the question.