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Cube Root of 392

The value of the cube root of 392 rounded to 6 decimal places is 7.318611. It is the real solution of the equation x³ = 392. The cube root of 392 is expressed as ∛392 or 2 ∛49 in the radical form and as (392)^⅓ or (392)^0.33 in the exponent form. The prime factorization of 392 is 2 × 2 × 2 × 7 × 7, hence, the cube root of 392 in its lowest radical form is expressed as 2 ∛49.

Cube root of 392: 7.31861142
Cube root of 392 in Exponential Form: (392)^⅓
Cube root of 392 in Radical Form: ∛392 or 2 ∛49

1.	What is the Cube Root of 392?
2.	How to Calculate the Cube Root of 392?
3.	Is the Cube Root of 392 Irrational?
4.	FAQs on Cube Root of 392

What is the Cube Root of 392?

The cube root of 392 is the number which when multiplied by itself three times gives the product as 392. Since 392 can be expressed as 2 × 2 × 2 × 7 × 7. Therefore, the cube root of 392 = ∛(2 × 2 × 2 × 7 × 7) = 7.3186.

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How to Calculate the Value of the Cube Root of 392?

Cube Root of 392 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 = 392
Let us assume x as 7
[∵ 7³ = 343 and 343 is the nearest perfect cube that is less than 392]
⇒ x = 7
Therefore,
∛392 = 7 (7³ + 2 × 392)/(2 × 7³ + 392)) = 7.32
⇒ ∛392 ≈ 7.32
Therefore, the cube root of 392 is 7.32 approximately.

Is the Cube Root of 392 Irrational?

Yes, because ∛392 = ∛(2 × 2 × 2 × 7 × 7) = 2 ∛49 and it cannot be expressed in the form of p/q where q ≠ 0. Therefore, the value of the cube root of 392 is an irrational number.

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Cube Root of 392 Solved Examples

Example 1: What is the value of ∛392 + ∛(-392)?

Solution:

The cube root of -392 is equal to the negative of the cube root of 392.
i.e. ∛-392 = -∛392

Therefore, ∛392 + ∛(-392) = ∛392 - ∛392 = 0
Example 2: The volume of a spherical ball is 392π in³. What is the radius of this ball?

Solution:

Volume of the spherical ball = 392π in³
= 4/3 × π × R³
⇒ R³ = 3/4 × 392
⇒ R = ∛(3/4 × 392) = ∛(3/4) × ∛392 = 0.90856 × 7.31861 (∵ ∛(3/4) = 0.90856 and ∛392 = 7.31861)
⇒ R = 6.6494 in³
Example 3: Given the volume of a cube is 392 in³. Find the length of the side of the cube.

Solution:

Volume of the Cube = 392 in³ = a³
⇒ a³ = 392
Cube rooting on both sides,
⇒ a = ∛392 in
Since the cube root of 392 is 7.32, therefore, the length of the side of the cube is 7.32 in.

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FAQs on Cube Root of 392

What is the Value of the Cube Root of 392?

We can express 392 as 2 × 2 × 2 × 7 × 7 i.e. ∛392 = ∛(2 × 2 × 2 × 7 × 7) = 7.31861. Therefore, the value of the cube root of 392 is 7.31861.

If the Cube Root of 392 is 7.32, Find the Value of ∛0.392.

Let us represent ∛0.392 in p/q form i.e. ∛(392/1000) = 7.32/10 = 0.73. Hence, the value of ∛0.392 = 0.73.

Why is the Value of the Cube Root of 392 Irrational?

The value of the cube root of 392 cannot be expressed in the form of p/q where q ≠ 0. Therefore, the number ∛392 is irrational.

Is 392 a Perfect Cube?

The number 392 on prime factorization gives 2 × 2 × 2 × 7 × 7. Here, the prime factor 7 is not in the power of 3. Therefore the cube root of 392 is irrational, hence 392 is not a perfect cube.

What is the Cube of the Cube Root of 392?

The cube of the cube root of 392 is the number 392 itself i.e. (∛392)³ = (392^1/3)³ = 392.

What is the Cube Root of -392?

The cube root of -392 is equal to the negative of the cube root of 392. Therefore, ∛-392 = -(∛392) = -(7.319) = -7.319.