Evaluate ${{32}^{\dfrac{3}{5}}}$ ?

Hint: To evaluate the given value we will use prime factorization method. Firstly we will find the prime factor of 32 and write them in simplified form by adding power of numbers with the same base. Then we will try to cancel out the fraction power given and then simplify it to get the desired answer.

Complete step-by-step solution:
We have to evaluate the following value:
 ${{32}^{\dfrac{3}{5}}}$…..$\left( 1 \right)$
We will use prime factorization to simplify the base value which is 32 as follows:
$\begin{align}
  & 2\left| \!{\underline {\,
  32 \,}} \right. \\
 & 2\left| \!{\underline {\,
  16 \,}} \right. \\
 & 2\left| \!{\underline {\,
  8 \,}} \right. \\
 & 2\left| \!{\underline {\,
  4 \,}} \right. \\
 & 2\left| \!{\underline {\,
  2 \,}} \right. \\
 & 1 \\
\end{align}$
So we can write equation (1) as below:
${{\left( 2\times 2\times 2\times 2\times 2 \right)}^{\dfrac{3}{5}}}$
Now we will add the power of factor with same base and simplify it as below:
$\begin{align}
  & \Rightarrow {{\left( {{2}^{5}} \right)}^{\dfrac{3}{5}}} \\
 & \Rightarrow {{2}^{5\times \dfrac{3}{5}}} \\
 & \Rightarrow {{2}^{3}} \\
 & \Rightarrow 8 \\
\end{align}$
So we get the value as 8.
Hence on evaluating ${{32}^{\dfrac{3}{5}}}$ we get the answer as 8.

Note: Prime factorization is a method to find the prime factor of any number which when multiplied gives that number. We use prime factorization to find out the HCF (Highest Common Factor) and LCM (Least Common Multiple) of any given set of numbers. This method is used for composite numbers and is not applicable on prime numbers. There are two ways to find the prime factors which are by Division method and Factor tree method. The division method is mostly used and is easy to understand in comparison to Factor Tree method. In the division method we divide the number with the prime numbers starting from the smallest prime number till we get a remainder as 0 and then all the prime numbers used in the process are noted down and are known as prime factors.