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.
• 81^1/3 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 = 3^4×1/3   as we know that (a^m)ⁿ = a^m×n

= 3^4/3 

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