Cube Root of 10

A real number whose cube is 10 is referred to as the cube root of 10. The opposite procedure of calculating the cube of an integer is finding its cube root. If “a” equals the cube root of 10 (i.e., a = ∛10), then a³ = 10. The radical sign “√” is commonly used to represent the square root of an integer, whereas “∛” represents the cube root. In exponential notation, the cube root of 10 is denoted by the expression 10^1/3. Let us see how to compute the cube root of 10 using several ways.

Also, read: How to Find Cube Root?

What is the Cube Root of 10?

The number whose cube is 10 is referred to as the cube root of that number. A perfect cube is what is known as the cube of an integer. 10 is not a perfect cube number since it cannot be expressed as the cube of any integer. This article will explain both the cube root of 10 and the technique for computing the cube root of 10.

Also, try out: Cube Root Calculator.

How to Find the Cube Root of 10?

Let’s now calculate the cube root of 10. Here, we’ll determine it using a variety of methods.

Finding Cube Root of 10 by Prime Factorisation Method

Below is a list of the steps involved in computing a number’s cube root using the prime factorization method.

• Calculate the prime factors of the given number.
• Make a group of three for each prime factor in the prime factorization.
• Preceding the multiplication of all the prime factors, we take that factor once for each group. No factor can be further simplified if it cannot be broken down into three parts.

Let’s try to employ a prime factorization method to determine the cube root of 10 step by step.

We know that the prime factorisation of 10 is 2 × 5.

As a result, this can be expressed as,

The cube root of 10 = ∛10 = ∛(2 × 5) = 2.154

The cube root of 10 is also irrational because ∛10 cannot be reduced further and is an irrational number.

Finding Cube Root of 10 by Approximation Method

To calculate the cube root of any number, the approximation approach known as Halley’s method is employed. Using Halley’s method, the cube root of 10 can be calculated as follows:

Halley Formula to Find a Cube Root:

Where

The number “a” stands for the cube root that needs to be calculated.

The estimated value, “x,” is frequently determined by taking the cube root of the nearest perfect cube.

In this case, a = 10

Assume that x = 2 and therefore, 2³ = 8 < 10

When these values are substituted in the above given formula, we get a result that is very near to the cube root of 10.

∛10 = 2[(2³ + 2 × 10)/(2 × 2³ + 10)]

= 2[(28)/(26)]

=2[1.076] ≈ 2.152, which is very near to the actual cube root of 10.

Therefore, Halley’s method produces a cube root of 10 that is roughly equal to 2.152.

Video Lesson on Finding Cube Roots

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Solved Examples on Cube Root of 10

Example 1:

Find the smallest integer that needs to be multiplied by 10 in order to produce a perfect cube number.

Solution:

We already know that the prime factorisation of 10 is 2 × 5.

We should group each factor by three in order to obtain the cube root of 10. Therefore, to obtain a cube number, we must multiply 10 by 2 × 2 × 5 × 5.

This means that 10 × 4 × 25 = 10 × 100 = 1000 is a perfect cube number.

The cube root of 1000 is, therefore ∛(2 × 2 × 2 × 5 × 5 × 5) = 2 × 5 = 10

Therefore, 100 is the smallest integer that must be multiplied by 10 to produce a perfect cube number.

Example 2:

Find the cube’s side length, if its volume is 10 cm³.

Solution:

Assume that “a” represents the cube’s side length.

Thus, the volume of the cube = x³ = 10 cm³

By taking the cube root on both sides of the equation, we get

⇒ x = ∛10 = 2.154 cm (approx.)

Example 3:

Find the value of x, if ∛10x = 2.154.

Solution:

Given equation: ∛10x = 2.154.

We know that the cube root of 10 is approximately equal to 2.154.

Now, substitute ∛10 = 2.154 in the given equation, we get

2.154 x = 2.154

⇒ x = 2.154 / 2.154

⇒ x = 1.

Hence, the value of x is 1.