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Cube Root of 189

The value of the cube root of 189 rounded to 5 decimal places is 5.73879. It is the real solution of the equation x³ = 189. The cube root of 189 is expressed as ∛189 or 3 ∛7 in the radical form and as (189)^⅓ or (189)^0.33 in the exponent form. The prime factorization of 189 is 3 × 3 × 3 × 7, hence, the cube root of 189 in its lowest radical form is expressed as 3 ∛7.

Cube root of 189: 5.738793548
Cube root of 189 in Exponential Form: (189)^⅓
Cube root of 189 in Radical Form: ∛189 or 3 ∛7

Cube Root of 189

1.	What is the Cube Root of 189?
2.	How to Calculate the Cube Root of 189?
3.	Is the Cube Root of 189 Irrational?
4.	FAQs on Cube Root of 189

What is the Cube Root of 189?

The cube root of 189 is the number which when multiplied by itself three times gives the product as 189. Since 189 can be expressed as 3 × 3 × 3 × 7. Therefore, the cube root of 189 = ∛(3 × 3 × 3 × 7) = 5.7388.

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How to Calculate the Value of the Cube Root of 189?

Cube Root of 189 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 = 189
Let us assume x as 5
[∵ 5³ = 125 and 125 is the nearest perfect cube that is less than 189]
⇒ x = 5
Therefore,
∛189 = 5 (5³ + 2 × 189)/(2 × 5³ + 189)) = 5.73
⇒ ∛189 ≈ 5.73
Therefore, the cube root of 189 is 5.73 approximately.

Is the Cube Root of 189 Irrational?

Yes, because ∛189 = ∛(3 × 3 × 3 × 7) = 3 ∛7 and it cannot be expressed in the form of p/q where q ≠ 0. Therefore, the value of the cube root of 189 is an irrational number.

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Cube Root of 189 Solved Examples

Example 1: What is the value of ∛189 + ∛(-189)?

Solution:

The cube root of -189 is equal to the negative of the cube root of 189.
i.e. ∛-189 = -∛189

Therefore, ∛189 + ∛(-189) = ∛189 - ∛189 = 0
Example 2: Find the real root of the equation x³ − 189 = 0.

Solution:

x³ − 189 = 0 i.e. x³ = 189
Solving for x gives us,
x = ∛189, x = ∛189 × (-1 + √3i))/2 and x = ∛189 × (-1 - √3i))/2
where i is called the imaginary unit and is equal to √-1.
Ignoring imaginary roots,
x = ∛189
Therefore, the real root of the equation x³ − 189 = 0 is for x = ∛189 = 5.7388.
Example 3: The volume of a spherical ball is 189π in³. What is the radius of this ball?

Solution:

Volume of the spherical ball = 189π in³
= 4/3 × π × R³
⇒ R³ = 3/4 × 189
⇒ R = ∛(3/4 × 189) = ∛(3/4) × ∛189 = 0.90856 × 5.73879 (∵ ∛(3/4) = 0.90856 and ∛189 = 5.73879)
⇒ R = 5.21404 in³

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FAQs on Cube Root of 189

What is the Value of the Cube Root of 189?

We can express 189 as 3 × 3 × 3 × 7 i.e. ∛189 = ∛(3 × 3 × 3 × 7) = 5.73879. Therefore, the value of the cube root of 189 is 5.73879.

What is the Value of 18 Plus 4 Cube Root 189?

The value of ∛189 is 5.739. So, 18 + 4 × ∛189 = 18 + 4 × 5.739 = 40.956. Hence, the value of 18 plus 4 cube root 189 is 40.956.

Is 189 a Perfect Cube?

The number 189 on prime factorization gives 3 × 3 × 3 × 7. Here, the prime factor 7 is not in the power of 3. Therefore the cube root of 189 is irrational, hence 189 is not a perfect cube.

If the Cube Root of 189 is 5.74, Find the Value of ∛0.189.

Let us represent ∛0.189 in p/q form i.e. ∛(189/1000) = 5.74/10 = 0.57. Hence, the value of ∛0.189 = 0.57.

What is the Cube of the Cube Root of 189?

The cube of the cube root of 189 is the number 189 itself i.e. (∛189)³ = (189^1/3)³ = 189.

What is the Cube Root of -189?

The cube root of -189 is equal to the negative of the cube root of 189. Therefore, ∛-189 = -(∛189) = -(5.739) = -5.739.

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