The fifth root of $32$ is $2$.

To comprehend this concept, let’s break it down step by step. The fifth root of a number is the value that, when multiplied by itself five times, results in the original number. In mathematical terms, if $x$ represents the fifth root of $32$, then we can express this relationship as:

To find $x$, we can start by examining the prime factorization of $32$. The number $32$ can be expressed as $2^5$, since:

This demonstrates that $32$ consists of five factors of $2$.

Now, when we take the fifth root of $2^5$, we are essentially asking, “What number, when raised to the power of $5$, equals $2^5$?” The answer is $2$, because:

In a more general sense, the $n$th root of $a^n$ is $a$. Therefore, the fifth root of $2^5$ is indeed $2$. This principle can be applied broadly to any similar problems involving roots and powers.

Understanding roots and powers is vital in GCSE Mathematics, as it aids in solving equations and grasping the properties of numbers. It’s important to remember that roots serve as the inverse operation to powers, just as subtraction is the inverse of addition.