Question

What are the fourth roots of $16$?

Answer 1

Step 1: Solve for fourth root of $16.$

The fourth root of $16$ is $\pm 2.$ A root is simply denoted by $\text{'}\sqrt{}\text{'}.$ Therefore, a fourth root will be denoted by $\sqrt[4]{}.$

Hence, we can write, fourth root of number $16$ as:

$\sqrt[4]{16}=\pm 2$

The fourth root of any number will give a value, which on multiplication by itself, four-times, will produce the original number.

Suppose, $n$ is the number and $x$ is the fourth root of $n$, then;

$\sqrt[4]{n}=x$

Or

$n=x^4=x.x.x.x$

Thus, in case of $\sqrt[4]{16}$, we have:

$\sqrt[4]{16}=\pm 2$

Or

$16=2^4$

Step 2: Fourth Root of $16$ by Prime Factorization.

We can also find the fourth root of $16$ by prime factorization method.

Since, prime factorization of $16$ is;

$16=2x2x2x2$

Therefore, if we take the fourth root on both sides, we get;

$$\begin{gathered} \sqrt[4]{16}=\sqrt[4]{\left(2\times 2\times 2\times 2\right)} \\ \sqrt[4]{16}=\pm 2 \end{gathered}$$

Hence the fourth root of $16$ is $\pm 2.$