Cube Root of 81

Last Updated : 07 Mar, 2024

Cube Root of 81 is approximately equal to 4.326. As 81 is not a perfect cube, it cannot be expressed as the cube of an integer. Cube Root of 81 can be represented as ∛81 or 3∛3 or 3^4/3.

In this article, we will learn to find the cube root of 81 by various methods and some solved examples related to the same.

Cube-Root-of-81

Cube Root Definition

A number’s cube root is its value after it has been multiplied twice, giving the original integer.

Some examples of Cube Root are:

The cube root of 8 is 2, since 2³= 8.
The cube root of 27 is 3, since 3³= 27.
The cube root of 64 is 4, since 4³= 64.
The cube root of 125 is 5, since 5³= 125.
The cube root of 216 is 6, since 6³= 216.

What is Cube Root of 81?

A Cube Root of 81 is a value that when multiplied by itself three times results in 81. Mathematically, if x is the cube root of 81, it can be represented as:

x³ = 81

⇒ x = (81)^1/3

⇒ x = 4.32

Thus, Cube root of 81 is almost equals to 4.32.

Cube Root Calculator

Cube root of any number can be calculated using the following calculator:

How to Find Cube Root of 81?

To find the cube root of 81 various methods can be used that are given below:

Halley’s Method
Prime Factorisation

Cube Root of 81 by Halley’s Method

Halley’s Method referred to as the tangent hyperbolas method or Halley’s rational formula, which is used as a root-finding procedure. The following formula can also be used to get the cube root of any given number:

∛a ≈ x × (x³ + 2a) / (2x³ + a)

Where,

The cube root of the desired number is represented by ∛a.
x is an initial guess for the cube root, usually the nearest integer less than or equal to the cube root of a.

We know that ∛81 = 3 so let’s substitute that value:

Here, a = 81 and Let’s solve x:

∛81 ≈ x × (x³+ 2 × 81) / (2 × x³ + 81)

∛3 ≈ x × (x³ + 162) / (2x³ + 81)

Now Here, we do cross multiplication;

3 (2x³+81) = x(x³ + 162)

6x³+243 = x⁴ + 162 x

x⁴ – 6x³ −162x + 243 = 0

Cube Root of 81 by Prime Factorisation

Cube Root of 81 can be calculated using the Prime Factorisation as follows:

Step 1: Find the Prime Factors of 81.

Thus, Prime Factors of 81 = 3 × 3 × 3 × 3

Step 2: Pair three of each factors from prime factorization.

81 = (3 × 3 × 3) × 3

Step 3: Simplify.

∛81 = ∛[(3 × 3 × 3) × 3]

⇒ ∛81 = ∛[(3 × 3 × 3) × 3] = 3∛3

Thus, the cube root of 81 is 3∛3 ≈ 4.33. It can also be expressed in exponent form, is 3^4/3.

Facts about 81

Some common facts about number 81 are:

Divisibility: 81 is divisible by 1, 3, 9, 27, and 81 itself.
Square of 9: 81 is the square of the integer 9. In mathematics, a square number is the result of multiplying an integer by itself.
Perfect Square: Being the square of 9, 81 is a perfect square.
Prime Factorization: The prime factorization of 81 is 3 × 3 × 3 × 3 i.e., 3⁴
Binary Representation: In binary, 81 is represented as 1010001.
Symbolism: In numerology, 81 symbolizes leadership, independence, and determination.

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Examples on Cube Root of 81

Example 1: Simplify ∛(81) × ∛(-81)

Solution:

We know,

∛(81) = 4.3267

∛(-81) = -4.3267

Hence, ∛(81) × ∛(-81) = 4.3267 × (-4.3267)

Example 2: Given the volume of a cube is 81 inches³. Find the length of the side of the cube.

Solution

Volume of the Cube = 81 inches ³ = a³

a³ = 81

a = ∛81 inches

The side length of the cube is 4.33 inches since the cube root of 81 is 4.33.

Cube Root of 81: Practice Problems

Problem 1: Verify whether 9 is the cube root of 81.

Problem 2: If x³ = 81, find the value of x.

Problem 3: Calculate the cube root of 81³.

Cube Root of 81: FAQs

What is the Value of the Cube Root of 81?

The cube root of 81 is 4.3267.

What is the cube root of 81 in radical form?

The cube root of 81 can be expressed as ∛(81) or 3∛3 in radical form.

Is 81 a Perfect Square?

Yes, 81 is a perfect square, as 9²= 81.

How many divisors 81 has?

81 has 5 divisors: 1, 3, 9, 27, and 81 itself.

What is Square of 81?

Value of square of 81 is 6561.

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