Cube Root of 11

The real number whose cube is 11 is the cube root of 11. 11 has a cube root of about 2.2239. Calculating a number’s cube is the reverse of finding its cube root. The radical marks “√” and “∛” respectively, are used to represent the square root and cube root. The cube root of 11 is represented exponentially by the notation 11^1/3. Let’s look at different approaches to find the cube root of 11.

Also, read: How to Find Cube Root?

What is the Cube Root of 11?

The number whose cube is 11 is known as the cube root of 11. A perfect cube is the cube of an integer. 11 is not a perfect cube number since it cannot be written as the cube of any integer. 11 is a prime number as well. The cube root of 11 and how to calculate it are discussed in this article.

Also, try out: Cube Root Calculator.

How to Find the Cube Root of 11?

Let’s now calculate the cube root of 11. We’ll cover it in many ways in this article.

Finding Cube Root of 11 by Prime Factorisation Method

The steps for computing a number’s cube root using the prime factorisation method are shown below.

• Determine the prime factors of the given integer.
• Make a group of three for each prime factor in the prime factorisation.
• Before multiplying all the prime factors, we take that factor only once for each group. A factor cannot be further simplified if it cannot be broken into three.

To obtain the cube root of 11, let’s continue to apply the prime factorisation method.

11 is known to have a prime factorisation of 1 × 11.

Hence, the cube root of 11 is equal to,

= ∛(1 × 11) = ∛11

The number 11 cannot be reduced even further. Hence its cube root is also an irrational number.

Finding Cube Root of 11 by Approximation Method

The cube root of any number may be calculated using Halley’s approach, sometimes referred to as an approximation method. Using Halley’s method, the cube root of 11 may be determined as follows:

Halley’s Cube Root Formula:

Where

The letter “a” designates the cube root that has to be calculated.

To get the estimated value, take the cube root of the closest perfect cube, “x”.

In this instance, a = 11

Assume x = 2, which results in, 2³ = 8 < 11

Halley’s formula gives us a result that is extremely close to the cube root of 11 when we plug in these numbers,

∛11 = 2[(2³ + 2 × 11)/(2 × 2³ + 11)]

= 2[(8 + 22)/(16 + 11)]

= 2[30/27]

= 2[1.11] ≈ 2.22, which is quite close to the cube root of 11.

Halley’s method yields a cube root of 11, which is 2.22.

Video Lesson on Finding Cube Roots

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Solved Examples on Cube Root of 11

Example 1:

If ∛11x = 4 (2.2239), then find the value of x.

Solution:

Given equation: ∛11x = 4 (2.2239).

The cube root of 11 is approximately 2.2239, as we already know.

when ∛11 = 2.2239 is replaced in the above equation, we get

2.2239 x = 4(2.2239)

⇒ x = [4(2.2239)] / 2.2239

⇒ x = 4.

Thus, x = 4.

Example 2:

Determine the smallest integer that can be multiplied by 11 to produce the perfect cube number.

Solution:

As it is well known, 1 × 11 is the prime factorisation of 11.

Each factor must be grouped by three in order to calculate the cube root of 11. Therefore, we need to multiply 11 by 11 × 11.

As a result, we get

11 × 11 × 11 = 11 × 121 = 1331

Hence, 1331 is a perfect cube number.

So, the cube root of 1331 is ∛(11 × 11 × 11 ) = 11.

So, 121 is the smallest number that can be multiplied by 11 to get a perfect cube number.

Example 3:

Calculate the length of a cube’s sides if its volume is 11 cm³.

Solution:

Consider “a” as the cube’s side length as a solution.

As a result, the cube’s volume is equal to a³ = 11 cm³.

When the cube roots are applied on both sides, we obtain

⇒ a = ∛11 = 2.2239 cm (approx).