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Cube Root of 65

The value of the cube root of 65 rounded to 4 decimal places is 4.0207. It is the real solution of the equation x³ = 65. The cube root of 65 is expressed as ∛65 in the radical form and as (65)^⅓ or (65)^0.33 in the exponent form. The prime factorization of 65 is 5 × 13, hence, the cube root of 65 in its lowest radical form is expressed as ∛65.

Cube root of 65: 4.020725759
Cube root of 65 in Exponential Form: (65)^⅓
Cube root of 65 in Radical Form: ∛65

1.	What is the Cube Root of 65?
2.	How to Calculate the Cube Root of 65?
3.	Is the Cube Root of 65 Irrational?
4.	FAQs on Cube Root of 65

What is the Cube Root of 65?

The cube root of 65 is the number which when multiplied by itself three times gives the product as 65. Since 65 can be expressed as 5 × 13. Therefore, the cube root of 65 = ∛(5 × 13) = 4.0207.

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How to Calculate the Value of the Cube Root of 65?

Cube Root of 65 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 = 65
Let us assume x as 4
[∵ 4³ = 64 and 64 is the nearest perfect cube that is less than 65]
⇒ x = 4
Therefore,
∛65 = 4 (4³ + 2 × 65)/(2 × 4³ + 65)) = 4.02
⇒ ∛65 ≈ 4.02
Therefore, the cube root of 65 is 4.02 approximately.

Is the Cube Root of 65 Irrational?

Yes, because ∛65 = ∛(5 × 13) and it cannot be expressed in the form of p/q where q ≠ 0. Therefore, the value of the cube root of 65 is an irrational number.

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Cube Root of 65 Solved Examples

Example 1: What is the value of ∛65 + ∛(-65)?

Solution:

The cube root of -65 is equal to the negative of the cube root of 65.
i.e. ∛-65 = -∛65

Therefore, ∛65 + ∛(-65) = ∛65 - ∛65 = 0
Example 2: Given the volume of a cube is 65 in³. Find the length of the side of the cube.

Solution:

Volume of the Cube = 65 in³ = a³
⇒ a³ = 65
Cube rooting on both sides,
⇒ a = ∛65 in
Since the cube root of 65 is 4.02, therefore, the length of the side of the cube is 4.02 in.
Example 3: The volume of a spherical ball is 65π in³. What is the radius of this ball?

Solution:

Volume of the spherical ball = 65π in³
= 4/3 × π × R³
⇒ R³ = 3/4 × 65
⇒ R = ∛(3/4 × 65) = ∛(3/4) × ∛65 = 0.90856 × 4.02073 (∵ ∛(3/4) = 0.90856 and ∛65 = 4.02073)
⇒ R = 3.65307 in³

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FAQs on Cube Root of 65

What is the Value of the Cube Root of 65?

We can express 65 as 5 × 13 i.e. ∛65 = ∛(5 × 13) = 4.02073. Therefore, the value of the cube root of 65 is 4.02073.

Is 65 a Perfect Cube?

The number 65 on prime factorization gives 5 × 13. Here, the prime factor 5 is not in the power of 3. Therefore the cube root of 65 is irrational, hence 65 is not a perfect cube.

What is the Value of 6 Plus 14 Cube Root 65?

The value of ∛65 is 4.021. So, 6 + 14 × ∛65 = 6 + 14 × 4.021 = 62.294. Hence, the value of 6 plus 14 cube root 65 is 62.294.

If the Cube Root of 65 is 4.02, Find the Value of ∛0.065.

Let us represent ∛0.065 in p/q form i.e. ∛(65/1000) = 4.02/10 = 0.4. Hence, the value of ∛0.065 = 0.4.

What is the Cube Root of -65?

The cube root of -65 is equal to the negative of the cube root of 65. Therefore, ∛-65 = -(∛65) = -(4.021) = -4.021.

What is the Cube of the Cube Root of 65?

The cube of the cube root of 65 is the number 65 itself i.e. (∛65)³ = (65^1/3)³ = 65.