# Cube Root of 81

The cube root of 81 is a number when multiplied by itself two times will be 81. The cube root of 81 is denoted by the symbol $\sqrt[3]{81}$. In this section, we will learn how to find the value of $\sqrt[3]{81}$. The value of the cube root of 81 is $3\sqrt[3]{3}$.

## Important Things about Cube Root of 81:

• $\sqrt[3]{81} = 3\sqrt[3]{3}$
• 81 is not a perfect cube.
• $\sqrt[3]{81}$ is the surd form of cube root of 81.
• 811/3 is the exponential form of cube root of 81.
• The cube root of 81 is 4.3267 in decimal form.
• cube of 81: 81×81×81 = 531441

## Definition of Cube Root of 81

Let m be a number such that m3=81. In this case, m is called a cube root of 81. Note that all cube roots of 81 are the solutions of the cubic equation x3=81. Let us now calculate the cube root of 81.

## What is the Cube Root of 81?

We will write 81 as a product of two numbers among which one of them will be a perfect cube. Note that 81=27×3. Here 27 is a perfect cube as 27=33. We have 81=27×3

Taking cube root on both sides, we have

$\sqrt[3]{81} = \sqrt[3]{27 \times 3}$ $=\sqrt[3]{27} \times \sqrt[3]{3}$ $[\because \sqrt[3]{x \times y}=\sqrt[3]{x} \times \sqrt[3]{y}]$

$=3 \times \sqrt[3]{3}$ as the cube root of 27 is 3.

$=3\sqrt[3]{3}$ So the cube root of 81 is $3\sqrt[3]{3}$.

This is the simplified form of the cube root of 81. As we know that $\sqrt[3]{3}=1.442$, the value of the cube root of 81 is equal to $3\sqrt[3]{3}=$ 3×1.442=4.326.

Express Cube root of 81 using indices: We know that 81=34. As cube root can be written as power 1/3, we have $\sqrt[3]{81}$ = (34)1/3 = 31/3   as we know that (am)n = am×n

= 34/3

This is the cube root of 81 in simplified exponent form.

## Cube root of 81 by Prime Factorization

To find the cube root of 81 by the prime factorization method, we will at first factorize 81. We know that  81=3×27, 27=3×9, and 9=3×3.

⇒ 81=3×27=3×3×9=3×3×3×3

So finally we get 81=3×3×3×3 …(∗)

Note that the above is the prime factorization of 81. Taking cube root on both sides of (∗), we obtain that

$\sqrt[3]{81} = \sqrt[3]{3 \times 3 \times 3 \times 3}$

$=\sqrt[3]{3 \times 3 \times 3} \times \sqrt[3]{3}$

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

∴ the cube root of 81 is $3 \sqrt[3]{3}$.

## Is 81 a perfect cube number?

We have calculated above that $\sqrt[3]{81}=3\sqrt[3]{3}$. Note that $\sqrt[3]{3}$ is not a whole number.

So $3\sqrt[3]{3}$ is also not a whole number, that is, the cube root of 81 is not a whole number.

Thus by the definition of a perfect cube, we conclude that 81 is not a perfect cube number.

## Is Cube Root of 81 Rational?

A number is said to be a rational number if it can be expressed as p/q where p and q are integers with q ≠ 0. From above notice that $\sqrt[3]{81}=3\sqrt[3]{3}$. As $\sqrt[3]{3}$ is not a rational number, the number $3\sqrt[3]{3}$ is also not a rational number. This makes the cube root of 81 an irrational number.

Conclusion: The cube root of 81 is not a rational number.

## Facts about 81

• 81 is a composite odd number.
• If a cube has a volume of 81 unit3, then we use the value of the cube root of 81 to find the length of the cube.
• A cube of length $3\sqrt[3]{3}$ unit has the volume 81 unit3

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