Fourth Root of 81

Let m be a number such that m4=81. In this case, m is called a fourth root of 81. The fourth roots of 81 are the solutions of x4=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 811/4 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=34

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 $81.$

Back to HOME PAGE