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Cube Root of 66

The value of the cube root of 66 rounded to 6 decimal places is 4.04124. It is the real solution of the equation x³ = 66. The cube root of 66 is expressed as ∛66 in the radical form and as (66)^⅓ or (66)^0.33 in the exponent form. The prime factorization of 66 is 2 × 3 × 11, hence, the cube root of 66 in its lowest radical form is expressed as ∛66.

Cube root of 66: 4.041240021
Cube root of 66 in Exponential Form: (66)^⅓
Cube root of 66 in Radical Form: ∛66

Cube Root of 66

1.	What is the Cube Root of 66?
2.	How to Calculate the Cube Root of 66?
3.	Is the Cube Root of 66 Irrational?
4.	FAQs on Cube Root of 66

What is the Cube Root of 66?

The cube root of 66 is the number which when multiplied by itself three times gives the product as 66. Since 66 can be expressed as 2 × 3 × 11. Therefore, the cube root of 66 = ∛(2 × 3 × 11) = 4.0412.

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How to Calculate the Value of the Cube Root of 66?

Cube Root of 66 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 = 66
Let us assume x as 4
[∵ 4³ = 64 and 64 is the nearest perfect cube that is less than 66]
⇒ x = 4
Therefore,
∛66 = 4 (4³ + 2 × 66)/(2 × 4³ + 66)) = 4.04
⇒ ∛66 ≈ 4.04
Therefore, the cube root of 66 is 4.04 approximately.

Is the Cube Root of 66 Irrational?

Yes, because ∛66 = ∛(2 × 3 × 11) and it cannot be expressed in the form of p/q where q ≠ 0. Therefore, the value of the cube root of 66 is an irrational number.

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Cube Root of 66 Solved Examples

Example 1: Given the volume of a cube is 66 in³. Find the length of the side of the cube.

Solution:

Volume of the Cube = 66 in³ = a³
⇒ a³ = 66
Cube rooting on both sides,
⇒ a = ∛66 in
Since the cube root of 66 is 4.04, therefore, the length of the side of the cube is 4.04 in.
Example 2: What is the value of ∛66 ÷ ∛(-66)?

Solution:

The cube root of -66 is equal to the negative of the cube root of 66.
⇒ ∛-66 = -∛66

Therefore,
⇒ ∛66/∛(-66) = ∛66/(-∛66) = -1
Example 3: Find the real root of the equation x³ − 66 = 0.

Solution:

x³ − 66 = 0 i.e. x³ = 66
Solving for x gives us,
x = ∛66, x = ∛66 × (-1 + √3i))/2 and x = ∛66 × (-1 - √3i))/2
where i is called the imaginary unit and is equal to √-1.
Ignoring imaginary roots,
x = ∛66
Therefore, the real root of the equation x³ − 66 = 0 is for x = ∛66 = 4.0412.

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FAQs on Cube Root of 66

What is the Value of the Cube Root of 66?

We can express 66 as 2 × 3 × 11 i.e. ∛66 = ∛(2 × 3 × 11) = 4.04124. Therefore, the value of the cube root of 66 is 4.04124.

What is the Cube Root of -66?

The cube root of -66 is equal to the negative of the cube root of 66. Therefore, ∛-66 = -(∛66) = -(4.041) = -4.041.

Is 66 a Perfect Cube?

The number 66 on prime factorization gives 2 × 3 × 11. Here, the prime factor 2 is not in the power of 3. Therefore the cube root of 66 is irrational, hence 66 is not a perfect cube.

If the Cube Root of 66 is 4.04, Find the Value of ∛0.066.

Let us represent ∛0.066 in p/q form i.e. ∛(66/1000) = 4.04/10 = 0.4. Hence, the value of ∛0.066 = 0.4.

Why is the Value of the Cube Root of 66 Irrational?

The value of the cube root of 66 cannot be expressed in the form of p/q where q ≠ 0. Therefore, the number ∛66 is irrational.

What is the Cube of the Cube Root of 66?

The cube of the cube root of 66 is the number 66 itself i.e. (∛66)³ = (66^1/3)³ = 66.

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