Cube Root of 81

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.