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Cube Root of 512

The value of the cube root of 512 is 8. It is the real solution of the equation x³ = 512. The cube root of 512 is expressed as ∛512 in radical form and as (512)^⅓ or (512)^0.33 in the exponent form. As the cube root of 512 is a whole number, 512 is a perfect cube.

Cube root of 512: 8
Cube root of 512 in exponential form: (512)^⅓
Cube root of 512 in radical form: ∛512

Cube Root of 512

1.	What is the Cube Root of 512?
2.	How to Calculate the Cube Root of 512?
3.	Is the Cube Root of 512 Irrational?
4.	FAQs on Cube Root of 512

What is the Cube Root of 512?

The cube root of 512 is the number which when multiplied by itself three times gives the product as 512. Since 512 can be expressed as 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2. Therefore, the cube root of 512 = ∛(2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2) = 8.

How to Calculate the Value of the Cube Root of 512?

Cube Root of 512 by Prime Factorization

Prime factorization of 512 is 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2
Simplifying the above expression: 2⁹
Simplifying further: 8³

Therefore, the cube root of 512 by prime factorization is (2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2)^1/3 = 8.

Is the Cube Root of 512 Irrational?

No, because ∛512 = ∛(2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2) can be expressed in the form of p/q i.e. 8/1. Therefore, the value of the cube root of 512 is an integer (rational).

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Cube Root of 512 Solved Examples

Example 1: The volume of a spherical ball is 512π in³. What is the radius of this ball?

Solution:

Volume of the spherical ball = 512π in³
= 4/3 × π × R³
⇒ R³ = 3/4 × 512
⇒ R = ∛(3/4 × 512) = ∛(3/4) × ∛512 = 0.90856 × 8 (∵ ∛(3/4) = 0.90856 and ∛512 = 8)
⇒ R = 7.26848 in³
Example 2: Given the volume of a cube is 512 in³. Find the length of the side of the cube.

Solution:

Volume of the Cube = 512 in³ = a³
⇒ a³ = 512
Cube rooting on both sides,
⇒ a = ∛512 in
Since the cube root of 512 is 8, therefore, the length of the side of the cube is 8 in.
Example 3: Find the real root of the equation x³ − 512 = 0.

Solution:

x³ − 512 = 0 i.e. x³ = 512
Solving for x gives us,
x = ∛512, x = ∛512 × (-1 + √3i))/2 and x = ∛512 × (-1 - √3i))/2
where i is called the imaginary unit and is equal to √-1.
Ignoring imaginary roots,
x = ∛512
Therefore, the real root of the equation x³ − 512 = 0 is for x = ∛512 = 8.

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FAQs on Cube Root of 512

What is the Value of the Cube Root of 512?

We can express 512 as 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 i.e. ∛512 = ∛(2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2) = 8. Therefore, the value of the cube root of 512 is 8.

Why is the value of the Cube Root of 512 Rational?

The value of the cube root of 512 can be expressed in the form of p/q i.e. = 8/1, where q ≠ 0. Therefore, the ∛512 is rational.

Is 512 a Perfect Cube?

The number 512 on prime factorization gives 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2. On combining the prime factors in groups of 3 gives 8. So, the cube root of 512 = ∛(2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2) = 8 (perfect cube).

What is the Cube Root of -512?

The cube root of -512 is equal to the negative of the cube root of 512. Therefore, ∛-512 = -(∛512) = -(8) = -8.

How to Simplify the Cube Root of 512/729?

We know that the cube root of 512 is 8 and the cube root of 729 is 9. Therefore, ∛(512/729) = (∛512)/(∛729) = 8/9 = 0.8889.

What is the Value of 7 Plus 1 Cube Root 512?

The value of ∛512 is 8. So, 7 + 1 × ∛512 = 7 + 1 × 8 = 15. Hence, the value of 7 plus 1 cube root 512 is 15.

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