Question

How do you find the fourth root of \[4096\]?

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 \[4096\], or we have to evaluate the value of \[\sqrt[4]{{4096}}\].
First, we find the prime factorization of \[4096\] and club the quadruplets of equal integers together.
\[\begin{array}{l}2\left| \!{\underline {\,
  {4096} \,}} \right. \\2\left| \!{\underline {\,
  {2048} \,}} \right. \\2\left| \!{\underline {\,
  {1024} \,}} \right. \\2\left| \!{\underline {\,
  {512} \,}} \right. \\2\left| \!{\underline {\,
  {256} \,}} \right. \\2\left| \!{\underline {\,
  {128} \,}} \right. \\2\left| \!{\underline {\,
  {64} \,}} \right. \\2\left| \!{\underline {\,
  {32} \,}} \right. \\2\left| \!{\underline {\,
  {16} \,}} \right. \\2\left| \!{\underline {\,
  8 \,}} \right. \\2\left| \!{\underline {\,
  4 \,}} \right. \\2\left| \!{\underline {\,
  2 \,}} \right. \\{\rm{ }}\left| \!{\underline {\,
  1 \,}} \right. \end{array}\]
Hence, \[96 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 = {2^4} \times {2^4} \times {2^4} = {8^4}\]
Hence, \[\sqrt[4]{{4096}} = \sqrt[4]{{{{\left( 8 \right)}^4}}} = 8\]
Thus, the fourth root of \[4096\] is \[8\].

Note: When we are calculating such questions, we find the prime factorization, club the quadruplets together, take them out as a single number and solve for it. This procedure requires no further action or steps to evaluate the answer. It is a point to remember that a perfect fourth power always has an even number of factors divisible by four.