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Cube Root of 180

The value of the cube root of 180 rounded to 4 decimal places is 5.6462. It is the real solution of the equation x³ = 180. The cube root of 180 is expressed as ∛180 in the radical form and as (180)^⅓ or (180)^0.33 in the exponent form. The prime factorization of 180 is 2 × 2 × 3 × 3 × 5, hence, the cube root of 180 in its lowest radical form is expressed as ∛180.

Cube root of 180: 5.646216173
Cube root of 180 in Exponential Form: (180)^⅓
Cube root of 180 in Radical Form: ∛180

Cube Root of 180

1.	What is the Cube Root of 180?
2.	How to Calculate the Cube Root of 180?
3.	Is the Cube Root of 180 Irrational?
4.	FAQs on Cube Root of 180

What is the Cube Root of 180?

The cube root of 180 is the number which when multiplied by itself three times gives the product as 180. Since 180 can be expressed as 2 × 2 × 3 × 3 × 5. Therefore, the cube root of 180 = ∛(2 × 2 × 3 × 3 × 5) = 5.6462.

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How to Calculate the Value of the Cube Root of 180?

Cube Root of 180 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 = 180
Let us assume x as 5
[∵ 5³ = 125 and 125 is the nearest perfect cube that is less than 180]
⇒ x = 5
Therefore,
∛180 = 5 (5³ + 2 × 180)/(2 × 5³ + 180)) = 5.64
⇒ ∛180 ≈ 5.64
Therefore, the cube root of 180 is 5.64 approximately.

Is the Cube Root of 180 Irrational?

Yes, because ∛180 = ∛(2 × 2 × 3 × 3 × 5) and it cannot be expressed in the form of p/q where q ≠ 0. Therefore, the value of the cube root of 180 is an irrational number.

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Cube Root of 180 Solved Examples

Example 1: The volume of a spherical ball is 180π in³. What is the radius of this ball?

Solution:

Volume of the spherical ball = 180π in³
= 4/3 × π × R³
⇒ R³ = 3/4 × 180
⇒ R = ∛(3/4 × 180) = ∛(3/4) × ∛180 = 0.90856 × 5.64622 (∵ ∛(3/4) = 0.90856 and ∛180 = 5.64622)
⇒ R = 5.12993 in³
Example 2: Given the volume of a cube is 180 in³. Find the length of the side of the cube.

Solution:

Volume of the Cube = 180 in³ = a³
⇒ a³ = 180
Cube rooting on both sides,
⇒ a = ∛180 in
Since the cube root of 180 is 5.65, therefore, the length of the side of the cube is 5.65 in.
Example 3: Find the real root of the equation x³ − 180 = 0.

Solution:

x³ − 180 = 0 i.e. x³ = 180
Solving for x gives us,
x = ∛180, x = ∛180 × (-1 + √3i))/2 and x = ∛180 × (-1 - √3i))/2
where i is called the imaginary unit and is equal to √-1.
Ignoring imaginary roots,
x = ∛180
Therefore, the real root of the equation x³ − 180 = 0 is for x = ∛180 = 5.6462.

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FAQs on Cube Root of 180

What is the Value of the Cube Root of 180?

We can express 180 as 2 × 2 × 3 × 3 × 5 i.e. ∛180 = ∛(2 × 2 × 3 × 3 × 5) = 5.64622. Therefore, the value of the cube root of 180 is 5.64622.

What is the Value of 10 Plus 14 Cube Root 180?

The value of ∛180 is 5.646. So, 10 + 14 × ∛180 = 10 + 14 × 5.646 = 89.044. Hence, the value of 10 plus 14 cube root 180 is 89.044.

How to Simplify the Cube Root of 180/64?

We know that the cube root of 180 is 5.64622 and the cube root of 64 is 4. Therefore, ∛(180/64) = (∛180)/(∛64) = 5.646/4 = 1.4115.

What is the Cube of the Cube Root of 180?

The cube of the cube root of 180 is the number 180 itself i.e. (∛180)³ = (180^1/3)³ = 180.

Is 180 a Perfect Cube?

The number 180 on prime factorization gives 2 × 2 × 3 × 3 × 5. Here, the prime factor 2 is not in the power of 3. Therefore the cube root of 180 is irrational, hence 180 is not a perfect cube.

If the Cube Root of 180 is 5.65, Find the Value of ∛0.18.

Let us represent ∛0.18 in p/q form i.e. ∛(180/1000) = 5.65/10 = 0.56. Hence, the value of ∛0.18 = 0.56.

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