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Cube Root of 4000

The value of the cube root of 4000 rounded to 5 decimal places is 15.87401. It is the real solution of the equation x³ = 4000. The cube root of 4000 is expressed as ∛4000 or 10 ∛4 in the radical form and as (4000)^⅓ or (4000)^0.33 in the exponent form. The prime factorization of 4000 is 2 × 2 × 2 × 2 × 2 × 5 × 5 × 5, hence, the cube root of 4000 in its lowest radical form is expressed as 10 ∛4.

Cube root of 4000: 15.87401052
Cube root of 4000 in Exponential Form: (4000)^⅓
Cube root of 4000 in Radical Form: ∛4000 or 10 ∛4

Cube Root of 4000

1.	What is the Cube Root of 4000?
2.	How to Calculate the Cube Root of 4000?
3.	Is the Cube Root of 4000 Irrational?
4.	FAQs on Cube Root of 4000

What is the Cube Root of 4000?

The cube root of 4000 is the number which when multiplied by itself three times gives the product as 4000. Since 4000 can be expressed as 2 × 2 × 2 × 2 × 2 × 5 × 5 × 5. Therefore, the cube root of 4000 = ∛(2 × 2 × 2 × 2 × 2 × 5 × 5 × 5) = 15.874.

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How to Calculate the Value of the Cube Root of 4000?

Cube Root of 4000 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 = 4000
Let us assume x as 15
[∵ 15³ = 3375 and 3375 is the nearest perfect cube that is less than 4000]
⇒ x = 15
Therefore,
∛4000 = 15 (15³ + 2 × 4000)/(2 × 15³ + 4000)) = 15.87
⇒ ∛4000 ≈ 15.87
Therefore, the cube root of 4000 is 15.87 approximately.

Is the Cube Root of 4000 Irrational?

Yes, because ∛4000 = ∛(2 × 2 × 2 × 2 × 2 × 5 × 5 × 5) = 10 ∛4 and it cannot be expressed in the form of p/q where q ≠ 0. Therefore, the value of the cube root of 4000 is an irrational number.

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Cube Root of 4000 Solved Examples

Example 1: The volume of a spherical ball is 4000π in³. What is the radius of this ball?

Solution:

Volume of the spherical ball = 4000π in³
= 4/3 × π × R³
⇒ R³ = 3/4 × 4000
⇒ R = ∛(3/4 × 4000) = ∛(3/4) × ∛4000 = 0.90856 × 15.87401 (∵ ∛(3/4) = 0.90856 and ∛4000 = 15.87401)
⇒ R = 14.42249 in³
Example 2: Given the volume of a cube is 4000 in³. Find the length of the side of the cube.

Solution:

Volume of the Cube = 4000 in³ = a³
⇒ a³ = 4000
Cube rooting on both sides,
⇒ a = ∛4000 in
Since the cube root of 4000 is 15.87, therefore, the length of the side of the cube is 15.87 in.
Example 3: Find the real root of the equation x³ − 4000 = 0.

Solution:

x³ − 4000 = 0 i.e. x³ = 4000
Solving for x gives us,
x = ∛4000, x = ∛4000 × (-1 + √3i))/2 and x = ∛4000 × (-1 - √3i))/2
where i is called the imaginary unit and is equal to √-1.
Ignoring imaginary roots,
x = ∛4000
Therefore, the real root of the equation x³ − 4000 = 0 is for x = ∛4000 = 15.874.

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FAQs on Cube Root of 4000

What is the Value of the Cube Root of 4000?

We can express 4000 as 2 × 2 × 2 × 2 × 2 × 5 × 5 × 5 i.e. ∛4000 = ∛(2 × 2 × 2 × 2 × 2 × 5 × 5 × 5) = 15.87401. Therefore, the value of the cube root of 4000 is 15.87401.

Is 4000 a Perfect Cube?

The number 4000 on prime factorization gives 2 × 2 × 2 × 2 × 2 × 5 × 5 × 5. Here, the prime factor 2 is not in the power of 3. Therefore the cube root of 4000 is irrational, hence 4000 is not a perfect cube.

How to Simplify the Cube Root of 4000/729?

We know that the cube root of 4000 is 15.87401 and the cube root of 729 is 9. Therefore, ∛(4000/729) = (∛4000)/(∛729) = 15.874/9 = 1.7638.

What is the Cube Root of -4000?

The cube root of -4000 is equal to the negative of the cube root of 4000. Therefore, ∛-4000 = -(∛4000) = -(15.874) = -15.874.

What is the Cube of the Cube Root of 4000?

The cube of the cube root of 4000 is the number 4000 itself i.e. (∛4000)³ = (4000^1/3)³ = 4000.

If the Cube Root of 4000 is 15.87, Find the Value of ∛4.

Let us represent ∛4.0 in p/q form i.e. ∛(4000/1000) = 15.87/10 = 1.59. Hence, the value of ∛4.0 = 1.59.

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