Cube Root of 243

A real number whose cube is 243 is known as the cube root of 243. The opposite operation of determining a number’s cube is finding its cube root. If a equals the cube root of 243 (i.e., a = ∛243), then a³ = 243. The symbol “√” is known as the radical sign and is often used to denote the square root of a number, whereas “∛” denotes the cube root. In exponential notation, 243’s cube root is represented as 243^1/3. Let’s see how to calculate the cube root of 243 using different methods.

Also, read: How to Find Cube Root?

What is the Cube Root of 243?

The number whose cube is 243 is known as the number’s cube root. The cube of an integer is what is known as a perfect cube. 243 is not a perfect cube number since we cannot define it in terms of the cube of any integer. The cube root of 243 and the method of computing the cube root of 243 are learned here.

Also, try out: Cube Root Calculator.

How to Find the Cube Root of 243?

Now let’s get the cube root of 243. Here, we’ll figure it out through several techniques.

Finding Cube Root of 243 by Prime Factorisation Method

The steps to compute a number’s cube root using the prime factorization technique are listed below.

• Identify the given number’s prime factors.
• In the prime factorization, create a group of 3 for each prime factor.
• We take that factor once for each group before multiplying all the prime factors. Any factor that cannot be divided into three cannot be further simplified.

Let’s try to calculate the cube root of 243 step by step using the prime factorisation method

As we know, the prime factorisation of 243 is 3 × 3 × 3 × 3 × 3.

Hence, this can be written as,

The cube root of 243 = ∛243 = ∛(3 × 3 × 3 × 3 × 3) = 3 × ∛9 = 3∛9.

Since ∛9 cannot be further reduced and ∛9 is an irrational number, 243’s cube root is also irrational.

Finding Cube Root of 243 by Approximation Method

The technique of approximation, commonly known as Halley’s method, is used to estimate the cube root of any number. The formula for calculating the cube root of 243 using Halley’s approach is as follows:

Halley Formula for Cube Root:

Where

“a” represents the integer whose cube root is to be calculated.

“x” is the estimated number, which is often calculated as the cube root of the closest perfect cube.

Here a = 243

Consider, x = 6 and hence, 6³ = 216 < 243

When these numbers are substituted into the preceding formula, we get,

∛243 = 6[(6³ + 2 × 243)/(2 × 6³ + 243)]

= 6[(702)/(675)]

= 6[1.04] ≈ 6.24, which is close to the actual cube root of 243.

Thus, the cube root of 243 using Halley’s method is approximately equal to 6.24.

Video Lesson on Finding Cube Roots

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• Cube Root Table
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• Cube Root of 729
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Solved Examples on Cube Root of 243

Example 1:

Determine the least integer that must be multiplied by 243 to get a perfect cube number.

Solution:

As we know, the prime factorisation of 243 is 3 × 3 × 3 × 3 × 3

To get the cube root of 243, we must group each factor by three. Consequently, we must multiply 243 by 3 to get a cube number.

I.e., 243 × 3 = 729 is a perfect cube number.

Thus, the cube root of 729 = ∛(3 × 3 × 3 × 3 × 3 × 3) = 3 × 3 = 9.

The least integer that needs to be multiplied by 243 to get a perfect cube number is, therefore, 3.

Example 2:

Determine the side length of a cube whose volume is 243 cm³.

Solution:

Let x represent the side of the cube.

We know that the volume of the cube = x³ = 243 cm³

⇒ x = ∛243 = 6.24 cm (approx.)

Example 3:

What is the least integer that must be divided from 243 to have a perfect cube number?

Solution:

As we know, the prime factorisation of 243 is 3 × 3 × 3 × 3 × 3.

Since 3 × 3 remains, and that cannot be grouped, 243 must be divided by 9.

I.e., 243 ÷ 9 = 27 is a perfect cube number.

The cube root of 27 = ∛(3 × 3 × 3) = 3.

Therefore, 9 is the least integer that must be divided from 243 to get a perfect cube number.