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Cube Root of 400

The value of the cube root of 400 rounded to 4 decimal places is 7.3681. It is the real solution of the equation x³ = 400. The cube root of 400 is expressed as ∛400 or 2 ∛50 in the radical form and as (400)^⅓ or (400)^0.33 in the exponent form. The prime factorization of 400 is 2 × 2 × 2 × 2 × 5 × 5, hence, the cube root of 400 in its lowest radical form is expressed as 2 ∛50.

Cube root of 400: 7.368062997
Cube root of 400 in Exponential Form: (400)^⅓
Cube root of 400 in Radical Form: ∛400 or 2 ∛50

Cube Root of 400

1.	What is the Cube Root of 400?
2.	How to Calculate the Cube Root of 400?
3.	Is the Cube Root of 400 Irrational?
4.	FAQs on Cube Root of 400

What is the Cube Root of 400?

The cube root of 400 is the number which when multiplied by itself three times gives the product as 400. Since 400 can be expressed as 2 × 2 × 2 × 2 × 5 × 5. Therefore, the cube root of 400 = ∛(2 × 2 × 2 × 2 × 5 × 5) = 7.3681.

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How to Calculate the Value of the Cube Root of 400?

Cube Root of 400 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 = 400
Let us assume x as 7
[∵ 7³ = 343 and 343 is the nearest perfect cube that is less than 400]
⇒ x = 7
Therefore,
∛400 = 7 (7³ + 2 × 400)/(2 × 7³ + 400)) = 7.37
⇒ ∛400 ≈ 7.37
Therefore, the cube root of 400 is 7.37 approximately.

Is the Cube Root of 400 Irrational?

Yes, because ∛400 = ∛(2 × 2 × 2 × 2 × 5 × 5) = 2 ∛50 and it cannot be expressed in the form of p/q where q ≠ 0. Therefore, the value of the cube root of 400 is an irrational number.

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Cube Root of 400 Solved Examples

Example 1: The volume of a spherical ball is 400π in³. What is the radius of this ball?

Solution:

Volume of the spherical ball = 400π in³
= 4/3 × π × R³
⇒ R³ = 3/4 × 400
⇒ R = ∛(3/4 × 400) = ∛(3/4) × ∛400 = 0.90856 × 7.36806 (∵ ∛(3/4) = 0.90856 and ∛400 = 7.36806)
⇒ R = 6.69432 in³
Example 2: Given the volume of a cube is 400 in³. Find the length of the side of the cube.

Solution:

Volume of the Cube = 400 in³ = a³
⇒ a³ = 400
Cube rooting on both sides,
⇒ a = ∛400 in
Since the cube root of 400 is 7.37, therefore, the length of the side of the cube is 7.37 in.
Example 3: Find the real root of the equation x³ − 400 = 0.

Solution:

x³ − 400 = 0 i.e. x³ = 400
Solving for x gives us,
x = ∛400, x = ∛400 × (-1 + √3i))/2 and x = ∛400 × (-1 - √3i))/2
where i is called the imaginary unit and is equal to √-1.
Ignoring imaginary roots,
x = ∛400
Therefore, the real root of the equation x³ − 400 = 0 is for x = ∛400 = 7.3681.

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FAQs on Cube Root of 400

What is the Value of the Cube Root of 400?

We can express 400 as 2 × 2 × 2 × 2 × 5 × 5 i.e. ∛400 = ∛(2 × 2 × 2 × 2 × 5 × 5) = 7.36806. Therefore, the value of the cube root of 400 is 7.36806.

How to Simplify the Cube Root of 400/729?

We know that the cube root of 400 is 7.36806 and the cube root of 729 is 9. Therefore, ∛(400/729) = (∛400)/(∛729) = 7.368/9 = 0.8187.

If the Cube Root of 400 is 7.37, Find the Value of ∛0.4.

Let us represent ∛0.4 in p/q form i.e. ∛(400/1000) = 7.37/10 = 0.74. Hence, the value of ∛0.4 = 0.74.

What is the Cube of the Cube Root of 400?

The cube of the cube root of 400 is the number 400 itself i.e. (∛400)³ = (400^1/3)³ = 400.

What is the Cube Root of -400?

The cube root of -400 is equal to the negative of the cube root of 400. Therefore, ∛-400 = -(∛400) = -(7.368) = -7.368.

Why is the Value of the Cube Root of 400 Irrational?

The value of the cube root of 400 cannot be expressed in the form of p/q where q ≠ 0. Therefore, the number ∛400 is irrational.

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