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Cube Root of 192

The value of the cube root of 192 rounded to 7 decimal places is 5.7689983. It is the real solution of the equation x³ = 192. The cube root of 192 is expressed as ∛192 or 4 ∛3 in the radical form and as (192)^⅓ or (192)^0.33 in the exponent form. The prime factorization of 192 is 2 × 2 × 2 × 2 × 2 × 2 × 3, hence, the cube root of 192 in its lowest radical form is expressed as 4 ∛3.

Cube root of 192: 5.768998281
Cube root of 192 in Exponential Form: (192)^⅓
Cube root of 192 in Radical Form: ∛192 or 4 ∛3

1.	What is the Cube Root of 192?
2.	How to Calculate the Cube Root of 192?
3.	Is the Cube Root of 192 Irrational?
4.	FAQs on Cube Root of 192

What is the Cube Root of 192?

The cube root of 192 is the number which when multiplied by itself three times gives the product as 192. Since 192 can be expressed as 2 × 2 × 2 × 2 × 2 × 2 × 3. Therefore, the cube root of 192 = ∛(2 × 2 × 2 × 2 × 2 × 2 × 3) = 5.769.

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How to Calculate the Value of the Cube Root of 192?

Cube Root of 192 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 = 192
Let us assume x as 5
[∵ 5³ = 125 and 125 is the nearest perfect cube that is less than 192]
⇒ x = 5
Therefore,
∛192 = 5 (5³ + 2 × 192)/(2 × 5³ + 192)) = 5.76
⇒ ∛192 ≈ 5.76
Therefore, the cube root of 192 is 5.76 approximately.

Is the Cube Root of 192 Irrational?

Yes, because ∛192 = ∛(2 × 2 × 2 × 2 × 2 × 2 × 3) = 4 ∛3 and it cannot be expressed in the form of p/q where q ≠ 0. Therefore, the value of the cube root of 192 is an irrational number.

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Cube Root of 192 Solved Examples

Example 1: What is the value of ∛192 ÷ ∛(-192)?

Solution:

The cube root of -192 is equal to the negative of the cube root of 192.
⇒ ∛-192 = -∛192

Therefore,
⇒ ∛192/∛(-192) = ∛192/(-∛192) = -1
Example 2: Given the volume of a cube is 192 in³. Find the length of the side of the cube.

Solution:

Volume of the Cube = 192 in³ = a³
⇒ a³ = 192
Cube rooting on both sides,
⇒ a = ∛192 in
Since the cube root of 192 is 5.77, therefore, the length of the side of the cube is 5.77 in.
Example 3: The volume of a spherical ball is 192π in³. What is the radius of this ball?

Solution:

Volume of the spherical ball = 192π in³
= 4/3 × π × R³
⇒ R³ = 3/4 × 192
⇒ R = ∛(3/4 × 192) = ∛(3/4) × ∛192 = 0.90856 × 5.769 (∵ ∛(3/4) = 0.90856 and ∛192 = 5.769)
⇒ R = 5.24148 in³

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FAQs on Cube Root of 192

What is the Value of the Cube Root of 192?

We can express 192 as 2 × 2 × 2 × 2 × 2 × 2 × 3 i.e. ∛192 = ∛(2 × 2 × 2 × 2 × 2 × 2 × 3) = 5.769. Therefore, the value of the cube root of 192 is 5.769.

If the Cube Root of 192 is 5.77, Find the Value of ∛0.192.

Let us represent ∛0.192 in p/q form i.e. ∛(192/1000) = 5.77/10 = 0.58. Hence, the value of ∛0.192 = 0.58.

How to Simplify the Cube Root of 192/216?

We know that the cube root of 192 is 5.769 and the cube root of 216 is 6. Therefore, ∛(192/216) = (∛192)/(∛216) = 5.769/6 = 0.9615.

What is the Cube Root of -192?

The cube root of -192 is equal to the negative of the cube root of 192. Therefore, ∛-192 = -(∛192) = -(5.769) = -5.769.

Is 192 a Perfect Cube?

The number 192 on prime factorization gives 2 × 2 × 2 × 2 × 2 × 2 × 3. Here, the prime factor 3 is not in the power of 3. Therefore the cube root of 192 is irrational, hence 192 is not a perfect cube.

What is the Cube of the Cube Root of 192?

The cube of the cube root of 192 is the number 192 itself i.e. (∛192)³ = (192^1/3)³ = 192.