Cube Root of 32

The cube root of 32 is a real number whose cube is 32. The cube root of 32 is about 3.1748. Determining a number’s cube root is the opposite of computing a number’s cube. The square root and cube root are typically denoted by the radical signs “√” and “∛”, respectively. The symbol 32^1/3 represents the exponential representation of the cube root of 32. Let’s examine various methods for determining the cube root of 32.

Also, read: How to Find Cube Root?

What is the Cube Root of 32?

The cube root of 32 is the number whose cube is 32. The cube of an integer is a perfect cube. Since 32 cannot be expressed as the cube of any integer, it is not a perfect cube number. In addition, 32 is a composite number. This article explains the cube root of 32 and how to compute it.

Also, try out: Cube Root Calculator.

How to Find the Cube Root of 32?

Now let’s determine the cube root of 32. Here, we’ll address it in a number of ways.

Finding Cube Root of 32 by Prime Factorisation Method

Below are the procedures for using the prime factorisation method to calculate a number’s cube root.

• Calculate the given number’s prime factors.
• For each prime factor in the prime factorisation, create a group of three.
• We take that factor once for each group before multiplying all the prime factors. If a factor cannot be divided into three, it cannot be further simplified.

Let’s proceed to use the prime factorisation technique to get the cube root of 32.

The prime factorisation of 32 is known to be 2 × 2 × 2 × 2 × 2.

The cube root of 32 is therefore equal to

= ∛(2 × 2 × 2 × 2 × 2) = 2 ∛4

Due to the fact that 2 ∛4 cannot be decreased further, its cube root is also an irrational number.

Finding Cube Root of 32 by Approximation Method

Halley’s method, often known as an approximation method, may be used to get the cube root of any integer. The cube root of 32 may be calculated using Halley’s approach as follows:

Cube Root Using the Halley Formula:

Where

The cube root that has to be computed is indicated by the letter “a.”

Using the cube root of the closest perfect cube, “x,” is used to get the estimated value.

In this case, a = 32

Suppose x = 3 and hence, we get, 3³ = 27 < 32

When we put these numbers into Halley’s formula, we get a value that is very near to the cube root of 32:

∛32 = 3[(3³ + 2 × 32)/(2 × 3³ + 32)]

= 3[(27 + 64)/(54 + 32)]

= 3[91/86]

= 3[1.058] ≈ 3.174

This result is quite similar to the cube root of 32.

Halley’s approach results in a cube root of 32, which is 3.174.

Video Lesson on Finding Cube Roots

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Solved Examples on Cube Root of 32

Example 1:

Determine the value of x, if ∛32x = 3.1748.

Solution:

The given equation is ∛32x = 3.1748.

We already know that 3.1748 is about equal to 32’s cube root.

When ∛32 = 3.1748 is substituted in the given equation, the result is,

3.1748 x = 3.1748

⇒ x = 3.1748 / 3.1748

⇒ x = 1.

Therefore, x = 1.

Example 2:

Find the least integer that must be multiplied by 32 to yield the perfect cube number.

Solution:

Since it is already known, the prime factorization of 32 = 2 × 2 × 2 × 2 × 2.

To find the cube root of 32, each element must be grouped by three. As a result, we must multiply 32 by 2.

This indicates that 64 is the perfect cube number (i.e., 32 × 2 = 64)

So, the cube root of 64 is ∛(2 × 2 × 2 × 2 × 2 × 2 ) = 2 × 2 = 4

Therefore, the least number that can be multiplied by 32 to get a perfect cube number is 2.

Example 3:

If a cube’s volume is 32 cm³, determine how long its sides should be.

Solution:

Take “a” as the cube’s side length.

The volume of the cube is therefore equal to a³ = 32 cm³.

Applying the cube roots on both sides, we get,

⇒ a = ∛32 = 3.1748 cm approximately.