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Cube Root of 384

The value of the cube root of 384 rounded to 5 decimal places is 7.26848. It is the real solution of the equation x³ = 384. The cube root of 384 is expressed as ∛384 or 4 ∛6 in the radical form and as (384)^⅓ or (384)^0.33 in the exponent form. The prime factorization of 384 is 2 × 2 × 2 × 2 × 2 × 2 × 2 × 3, hence, the cube root of 384 in its lowest radical form is expressed as 4 ∛6.

Cube root of 384: 7.268482371
Cube root of 384 in Exponential Form: (384)^⅓
Cube root of 384 in Radical Form: ∛384 or 4 ∛6

Cube Root of 384

1.	What is the Cube Root of 384?
2.	How to Calculate the Cube Root of 384?
3.	Is the Cube Root of 384 Irrational?
4.	FAQs on Cube Root of 384

What is the Cube Root of 384?

The cube root of 384 is the number which when multiplied by itself three times gives the product as 384. Since 384 can be expressed as 2 × 2 × 2 × 2 × 2 × 2 × 2 × 3. Therefore, the cube root of 384 = ∛(2 × 2 × 2 × 2 × 2 × 2 × 2 × 3) = 7.2685.

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How to Calculate the Value of the Cube Root of 384?

Cube Root of 384 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 = 384
Let us assume x as 7
[∵ 7³ = 343 and 343 is the nearest perfect cube that is less than 384]
⇒ x = 7
Therefore,
∛384 = 7 (7³ + 2 × 384)/(2 × 7³ + 384)) = 7.27
⇒ ∛384 ≈ 7.27
Therefore, the cube root of 384 is 7.27 approximately.

Is the Cube Root of 384 Irrational?

Yes, because ∛384 = ∛(2 × 2 × 2 × 2 × 2 × 2 × 2 × 3) = 4 ∛6 and it cannot be expressed in the form of p/q where q ≠ 0. Therefore, the value of the cube root of 384 is an irrational number.

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Cube Root of 384 Solved Examples

Example 1: The volume of a spherical ball is 384π in³. What is the radius of this ball?

Solution:

Volume of the spherical ball = 384π in³
= 4/3 × π × R³
⇒ R³ = 3/4 × 384
⇒ R = ∛(3/4 × 384) = ∛(3/4) × ∛384 = 0.90856 × 7.26848 (∵ ∛(3/4) = 0.90856 and ∛384 = 7.26848)
⇒ R = 6.60385 in³
Example 2: What is the value of ∛384 ÷ ∛(-384)?

Solution:

The cube root of -384 is equal to the negative of the cube root of 384.
⇒ ∛-384 = -∛384

Therefore,
⇒ ∛384/∛(-384) = ∛384/(-∛384) = -1
Example 3: Find the real root of the equation x³ − 384 = 0.

Solution:

x³ − 384 = 0 i.e. x³ = 384
Solving for x gives us,
x = ∛384, x = ∛384 × (-1 + √3i))/2 and x = ∛384 × (-1 - √3i))/2
where i is called the imaginary unit and is equal to √-1.
Ignoring imaginary roots,
x = ∛384
Therefore, the real root of the equation x³ − 384 = 0 is for x = ∛384 = 7.2685.

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FAQs on Cube Root of 384

What is the Value of the Cube Root of 384?

We can express 384 as 2 × 2 × 2 × 2 × 2 × 2 × 2 × 3 i.e. ∛384 = ∛(2 × 2 × 2 × 2 × 2 × 2 × 2 × 3) = 7.26848. Therefore, the value of the cube root of 384 is 7.26848.

How to Simplify the Cube Root of 384/512?

We know that the cube root of 384 is 7.26848 and the cube root of 512 is 8. Therefore, ∛(384/512) = (∛384)/(∛512) = 7.268/8 = 0.9085.

What is the Cube of the Cube Root of 384?

The cube of the cube root of 384 is the number 384 itself i.e. (∛384)³ = (384^1/3)³ = 384.

What is the Cube Root of -384?

The cube root of -384 is equal to the negative of the cube root of 384. Therefore, ∛-384 = -(∛384) = -(7.268) = -7.268.

Is 384 a Perfect Cube?

The number 384 on prime factorization gives 2 × 2 × 2 × 2 × 2 × 2 × 2 × 3. Here, the prime factor 2 is not in the power of 3. Therefore the cube root of 384 is irrational, hence 384 is not a perfect cube.

If the Cube Root of 384 is 7.27, Find the Value of ∛0.384.

Let us represent ∛0.384 in p/q form i.e. ∛(384/1000) = 7.27/10 = 0.73. Hence, the value of ∛0.384 = 0.73.

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