Question

What is the principal fifth root of \[32\] ?

Answer 1

Hint: Before solving this problem, we should have a clear understanding of indices and roots. Given a real number “a”, its principal nth root is the unique real solution of \[{{x}^{n}}=a\] . Now as per the given problem, we need to find the principal fifth root of \[32\] , or we need to calculate the unique real solution of \[{{x}^{5}}=32\] . Using prime factorization of \[32\] , we get that it breaks down into five number of twos. Hence, we can conclude that the fifth principal root of \[32\] gives \[2\] .

Complete step-by-step solution:
Indices are a representation of repetitive multiplication of the same kind. Repetitive multiplication if shown in the form $a\times a\times a\times a\times ....$ will become tedious and time taking. So, in order to avoid it, we have implemented the indices representation. Here, we represent the original number with the number of multiplications written as a superscript. For example, three times multiplication of two will be $2\times 2\times 2$ which can be written as ${{2}^{3}}$ . Indices can be called as an operation of numbers. The inverse operation of indices is called square rooting, cube rooting and so on. nth rooting means to break down a number into n other similar numbers such that their product gives the original number. For example, the cube root of $8$ gives $2$ since $2\times 2\times 2$ implies $8$ .
In nth rooting, we use prime factorisation to break down a number into its prime factors. Prime factorisation gives the product of prime factors. For example, the prime factorisation of $32$ gives,
\[\begin{align}
  & 2\left| \!{\underline {\,
  32 \,}} \right. \\
 & 2\left| \!{\underline {\,
  16 \,}} \right. \\
 & 2\left| \!{\underline {\,
  8 \,}} \right. \\
 & 2\left| \!{\underline {\,
  4 \,}} \right. \\
 & 2\left| \!{\underline {\,
  2 \,}} \right. \\
 & ~~~1 \\
\end{align}\]
Which can be written as \[2\times 2\times 2\times 2\times 2={{2}^{5}}\] . After fifth rooting, we clearly get $2$ .
Hence, we can conclude that the fifth principal root of $32$ gives $2$ .

Note: Problems including nth rooting of numbers can be easily done but there are chances of mistakes which should be avoided. We need to carefully perform the prime factorization of the numbers and check whether we get exactly “n” similar numbers for evaluating the correct answer.