Let m be a number such that m^{4}=81. In this case, m is called a fourth root of 81. The fourth roots of 81 are the solutions of x^{4}=81. It has four solutions since it has degree four. In this article, we will learn about the fourth roots of eighty-one.

**• **$\sqrt[4]{81}$ is the radical form of the fourth root of 81 with index 4.

**• **Fourth root of 81 can be written as 81^{1/4 }according to the rule of indices. That is,

\[\sqrt[4]{81}=81^{1/4}\]

**What is the fourth root of 81?**

Note that 81 can be expressed as a product of four numbers of 3‘s, that is,

81=3×3×3×3

⇒ 81=3^{4}

Taking fourth root on both sides, we get that

$\sqrt[4]{81}=\sqrt[4]{3^4}$

As the fourth root is the same as the power 1/4, we have

$\sqrt[4]{81}=(3^4)^{1/4}$

$\Rightarrow \sqrt[4]{81}=3^{4 \times 1/4}$ $[\because (a^m)^n=a^{m \times n}]$

$\Rightarrow \sqrt[4]{81}=3^{1}=3$

So the value of the fourth root of 81 is 3. This is one of the real fourth roots of 81.

**Is Fourth Root of 81 Rational?**

Since the fourth root of 81 is equal to 3 and we know that 3 is a rational number, then we conclude that the fourth root of 81 is a rational number; in fact, it is a whole number (or an integer).

For the above same reason, 81 is a perfect fourth number.

**How to Find ****Fourth Root of 81?**

Now, we will find the fourth root of 81 by the prime factorization method. Observe that

81=3×27, 27=3×9 and 9=3×3

Thus, 81 = 3×27 = 3×3×9 = 3×3×3×3.

So finally we get that

81 = 3×3×3×3

$\therefore \sqrt[4]{81}=\sqrt[4]{3\times 3 \times 3 \times 3}=3$ as we know that $\sqrt[4]{a \times a \times a \times a}=a$

So the fourth root of 81 is 3.

Question: What is the fourth root of 81?

Video Solution:

**Complex Fourth Roots of 81**

We will find the solutions of $x^4=81$ to find all the complex fourth roots of 81. We have:

$x^4=81 \cdot 1$

$\therefore x=81^{1/4} \cdot 1^{1/4}$ $\cdots (*)$

$\Rightarrow x=3 \cdot 1^{1/4}.$,

That is, $x$ is equal to $3$ times the fourth roots of $1.$

We know that there are $4$ fourth roots of $1$ and they are $\pm 1, \pm i$. Here $i$ is a complex imaginary number.

So from $(*)$, we obtain that the fourth roots of $81$ are $\pm 3$ and $\pm 3i.$ Here $\pm 3$ are the real roots and $\pm 3i$ are the complex roots of $x^4-81=0.$ Thus there are $2$ reals roots and $2$ complex roots of the equation $x^4=81.$ All these are the fourth roots of $81.$