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Cube Root of 324

The value of the cube root of 324 rounded to 7 decimal places is 6.8682855. It is the real solution of the equation x³ = 324. The cube root of 324 is expressed as ∛324 or 3 ∛12 in the radical form and as (324)^⅓ or (324)^0.33 in the exponent form. The prime factorization of 324 is 2 × 2 × 3 × 3 × 3 × 3, hence, the cube root of 324 in its lowest radical form is expressed as 3 ∛12.

Cube root of 324: 6.868285455
Cube root of 324 in Exponential Form: (324)^⅓
Cube root of 324 in Radical Form: ∛324 or 3 ∛12

Cube Root of 324

1.	What is the Cube Root of 324?
2.	How to Calculate the Cube Root of 324?
3.	Is the Cube Root of 324 Irrational?
4.	FAQs on Cube Root of 324

What is the Cube Root of 324?

The cube root of 324 is the number which when multiplied by itself three times gives the product as 324. Since 324 can be expressed as 2 × 2 × 3 × 3 × 3 × 3. Therefore, the cube root of 324 = ∛(2 × 2 × 3 × 3 × 3 × 3) = 6.8683.

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How to Calculate the Value of the Cube Root of 324?

Cube Root of 324 by Halley's Method

Its formula is ∛a ≈ x ((x³ + 2a)/(2x³ + a))
where,
a = number whose cube root is being calculated
x = integer guess of its cube root.

Here a = 324
Let us assume x as 6
[∵ 6³ = 216 and 216 is the nearest perfect cube that is less than 324]
⇒ x = 6
Therefore,
∛324 = 6 (6³ + 2 × 324)/(2 × 6³ + 324)) = 6.86
⇒ ∛324 ≈ 6.86
Therefore, the cube root of 324 is 6.86 approximately.

Is the Cube Root of 324 Irrational?

Yes, because ∛324 = ∛(2 × 2 × 3 × 3 × 3 × 3) = 3 ∛12 and it cannot be expressed in the form of p/q where q ≠ 0. Therefore, the value of the cube root of 324 is an irrational number.

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Cube Root of 324 Solved Examples

Example 1: Given the volume of a cube is 324 in³. Find the length of the side of the cube.

Solution:

Volume of the Cube = 324 in³ = a³
⇒ a³ = 324
Cube rooting on both sides,
⇒ a = ∛324 in
Since the cube root of 324 is 6.87, therefore, the length of the side of the cube is 6.87 in.
Example 2: The volume of a spherical ball is 324π in³. What is the radius of this ball?

Solution:

Volume of the spherical ball = 324π in³
= 4/3 × π × R³
⇒ R³ = 3/4 × 324
⇒ R = ∛(3/4 × 324) = ∛(3/4) × ∛324 = 0.90856 × 6.86829 (∵ ∛(3/4) = 0.90856 and ∛324 = 6.86829)
⇒ R = 6.24025 in³
Example 3: Find the real root of the equation x³ − 324 = 0.

Solution:

x³ − 324 = 0 i.e. x³ = 324
Solving for x gives us,
x = ∛324, x = ∛324 × (-1 + √3i))/2 and x = ∛324 × (-1 - √3i))/2
where i is called the imaginary unit and is equal to √-1.
Ignoring imaginary roots,
x = ∛324
Therefore, the real root of the equation x³ − 324 = 0 is for x = ∛324 = 6.8683.

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FAQs on Cube Root of 324

What is the Value of the Cube Root of 324?

We can express 324 as 2 × 2 × 3 × 3 × 3 × 3 i.e. ∛324 = ∛(2 × 2 × 3 × 3 × 3 × 3) = 6.86829. Therefore, the value of the cube root of 324 is 6.86829.

If the Cube Root of 324 is 6.87, Find the Value of ∛0.324.

Let us represent ∛0.324 in p/q form i.e. ∛(324/1000) = 6.87/10 = 0.69. Hence, the value of ∛0.324 = 0.69.

How to Simplify the Cube Root of 324/125?

We know that the cube root of 324 is 6.86829 and the cube root of 125 is 5. Therefore, ∛(324/125) = (∛324)/(∛125) = 6.868/5 = 1.3736.

What is the Cube of the Cube Root of 324?

The cube of the cube root of 324 is the number 324 itself i.e. (∛324)³ = (324^1/3)³ = 324.

Why is the Value of the Cube Root of 324 Irrational?

The value of the cube root of 324 cannot be expressed in the form of p/q where q ≠ 0. Therefore, the number ∛324 is irrational.

What is the Cube Root of -324?

The cube root of -324 is equal to the negative of the cube root of 324. Therefore, ∛-324 = -(∛324) = -(6.868) = -6.868.

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