Question

How do you find the fourth root of \[16\]?

Answer 1

Hint: In the given question, we have been asked to find the fourth root of an even natural number. To solve this question, we just need to know how to solve the fourth root. If the number is a perfect square, then it will have no integer left in the fourth root. But if it is not a perfect square, then it has at least one integer in the fourth root.

Complete step by step answer:
The given number whose simplified form is to be found is the fourth root of \[16\], or we have to evaluate the value of \[\sqrt[4]{{16}}\].
First, we find the prime factorization of \[16\] and club the quadruplets of equal integers together.
\[\begin{array}{l}2\left| \!{\underline {\,
  {16} \,}} \right. \\2\left| \!{\underline {\,
  8 \,}} \right. \\2\left| \!{\underline {\,
  4 \,}} \right. \\2\left| \!{\underline {\,
  2 \,}} \right. \\{\rm{ }}\left| \!{\underline {\,
  1 \,}} \right. \end{array}\]
Hence, \[16 = 2 \times 2 \times 2 \times 2 = {2^4}\]
Hence, \[\sqrt[4]{{16}} = \sqrt[4]{{{{\left( 2 \right)}^4}}} = 2\]

Thus, the fourth root of \[16\] is \[2\].

Note: In the given question, we had to find the value of the fourth root of a number. The questions of such type are pretty straight-forward. The methodology of solving is specific – factorize the number into prime factors, for each four equal numbers, club them as one and then multiply all the clubbed numbers. Students make mistakes while clubbing the numbers – would club more than four or less than four numbers or would make mistakes while multiplying the numbers. So, care must be taken at that point.