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Cube Root of 51

The value of the cube root of 51 rounded to 7 decimal places is 3.7084298. It is the real solution of the equation x³ = 51. The cube root of 51 is expressed as ∛51 in the radical form and as (51)^⅓ or (51)^0.33 in the exponent form. The prime factorization of 51 is 3 × 17, hence, the cube root of 51 in its lowest radical form is expressed as ∛51.

Cube root of 51: 3.708429769
Cube root of 51 in Exponential Form: (51)^⅓
Cube root of 51 in Radical Form: ∛51

1.	What is the Cube Root of 51?
2.	How to Calculate the Cube Root of 51?
3.	Is the Cube Root of 51 Irrational?
4.	FAQs on Cube Root of 51

What is the Cube Root of 51?

The cube root of 51 is the number which when multiplied by itself three times gives the product as 51. Since 51 can be expressed as 3 × 17. Therefore, the cube root of 51 = ∛(3 × 17) = 3.7084.

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How to Calculate the Value of the Cube Root of 51?

Cube Root of 51 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 = 51
Let us assume x as 3
[∵ 3³ = 27 and 27 is the nearest perfect cube that is less than 51]
⇒ x = 3
Therefore,
∛51 = 3 (3³ + 2 × 51)/(2 × 3³ + 51)) = 3.69
⇒ ∛51 ≈ 3.69
Therefore, the cube root of 51 is 3.69 approximately.

Is the Cube Root of 51 Irrational?

Yes, because ∛51 = ∛(3 × 17) and it cannot be expressed in the form of p/q where q ≠ 0. Therefore, the value of the cube root of 51 is an irrational number.

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Cube Root of 51 Solved Examples

Example 1: What is the value of ∛51 + ∛(-51)?

Solution:

The cube root of -51 is equal to the negative of the cube root of 51.
i.e. ∛-51 = -∛51

Therefore, ∛51 + ∛(-51) = ∛51 - ∛51 = 0
Example 2: The volume of a spherical ball is 51π in³. What is the radius of this ball?

Solution:

Volume of the spherical ball = 51π in³
= 4/3 × π × R³
⇒ R³ = 3/4 × 51
⇒ R = ∛(3/4 × 51) = ∛(3/4) × ∛51 = 0.90856 × 3.70843 (∵ ∛(3/4) = 0.90856 and ∛51 = 3.70843)
⇒ R = 3.36933 in³
Example 3: Find the real root of the equation x³ − 51 = 0.

Solution:

x³ − 51 = 0 i.e. x³ = 51
Solving for x gives us,
x = ∛51, x = ∛51 × (-1 + √3i))/2 and x = ∛51 × (-1 - √3i))/2
where i is called the imaginary unit and is equal to √-1.
Ignoring imaginary roots,
x = ∛51
Therefore, the real root of the equation x³ − 51 = 0 is for x = ∛51 = 3.7084.

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FAQs on Cube Root of 51

What is the Value of the Cube Root of 51?

We can express 51 as 3 × 17 i.e. ∛51 = ∛(3 × 17) = 3.70843. Therefore, the value of the cube root of 51 is 3.70843.

How to Simplify the Cube Root of 51/64?

We know that the cube root of 51 is 3.70843 and the cube root of 64 is 4. Therefore, ∛(51/64) = (∛51)/(∛64) = 3.708/4 = 0.927.

Why is the Value of the Cube Root of 51 Irrational?

The value of the cube root of 51 cannot be expressed in the form of p/q where q ≠ 0. Therefore, the number ∛51 is irrational.

Is 51 a Perfect Cube?

The number 51 on prime factorization gives 3 × 17. Here, the prime factor 3 is not in the power of 3. Therefore the cube root of 51 is irrational, hence 51 is not a perfect cube.

What is the Cube Root of -51?

The cube root of -51 is equal to the negative of the cube root of 51. Therefore, ∛-51 = -(∛51) = -(3.708) = -3.708.

What is the Value of 20 Plus 1 Cube Root 51?

The value of ∛51 is 3.708. So, 20 + 1 × ∛51 = 20 + 1 × 3.708 = 23.708. Hence, the value of 20 plus 1 cube root 51 is 23.708.