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Cube Root of 37

The value of the cube root of 37 rounded to 7 decimal places is 3.3322219. It is the real solution of the equation x³ = 37. The cube root of 37 is expressed as ∛37 in the radical form and as (37)^⅓ or (37)^0.33 in the exponent form. The prime factorization of 37 is 37, hence, the cube root of 37 in its lowest radical form is expressed as ∛37.

Cube root of 37: 3.332221852
Cube root of 37 in Exponential Form: (37)^⅓
Cube root of 37 in Radical Form: ∛37

Cube Root of 37

1.	What is the Cube Root of 37?
2.	How to Calculate the Cube Root of 37?
3.	Is the Cube Root of 37 Irrational?
4.	FAQs on Cube Root of 37

What is the Cube Root of 37?

The cube root of 37 is the number which when multiplied by itself three times gives the product as 37. The number 37 is prime. Therefore, the cube root of 37 = ∛37 = 3.3322.

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How to Calculate the Value of the Cube Root of 37?

Cube Root of 37 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 = 37
Let us assume x as 3
[∵ 3³ = 27 and 27 is the nearest perfect cube that is less than 37]
⇒ x = 3
Therefore,
∛37 = 3 (3³ + 2 × 37)/(2 × 3³ + 37)) = 3.33
⇒ ∛37 ≈ 3.33
Therefore, the cube root of 37 is 3.33 approximately.

Is the Cube Root of 37 Irrational?

Yes, because ∛37 cannot be expressed in the form of p/q where q ≠ 0. Therefore, the value of the cube root of 37 is an irrational number.

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Cube Root of 37 Solved Examples

Example 1: Given the volume of a cube is 37 in³. Find the length of the side of the cube.

Solution:

Volume of the Cube = 37 in³ = a³
⇒ a³ = 37
Cube rooting on both sides,
⇒ a = ∛37 in
Since the cube root of 37 is 3.33, therefore, the length of the side of the cube is 3.33 in.
Example 2: Find the real root of the equation x³ − 37 = 0.

Solution:

x³ − 37 = 0 i.e. x³ = 37
Solving for x gives us,
x = ∛37, x = ∛37 × (-1 + √3i))/2 and x = ∛37 × (-1 - √3i))/2
where i is called the imaginary unit and is equal to √-1.
Ignoring imaginary roots,
x = ∛37
Therefore, the real root of the equation x³ − 37 = 0 is for x = ∛37 = 3.3322.
Example 3: The volume of a spherical ball is 37π in³. What is the radius of this ball?

Solution:

Volume of the spherical ball = 37π in³
= 4/3 × π × R³
⇒ R³ = 3/4 × 37
⇒ R = ∛(3/4 × 37) = ∛(3/4) × ∛37 = 0.90856 × 3.33222 (∵ ∛(3/4) = 0.90856 and ∛37 = 3.33222)
⇒ R = 3.02752 in³

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FAQs on Cube Root of 37

What is the Value of the Cube Root of 37?

The value of the cube root of 37 is 3.33222.

What is the Value of 7 Plus 8 Cube Root 37?

The value of ∛37 is 3.332. So, 7 + 8 × ∛37 = 7 + 8 × 3.332 = 33.656. Hence, the value of 7 plus 8 cube root 37 is 33.656.

What is the Cube of the Cube Root of 37?

The cube of the cube root of 37 is the number 37 itself i.e. (∛37)³ = (37^1/3)³ = 37.

What is the Cube Root of -37?

The cube root of -37 is equal to the negative of the cube root of 37. Therefore, ∛-37 = -(∛37) = -(3.332) = -3.332.

Is 37 a Perfect Cube?

The number 37 is prime. Here, the prime factor 37 is not in the power of 3 and this implies that the cube root of 37 is irrational, hence 37 is not a perfect cube.

How to Simplify the Cube Root of 37/512?

We know that the cube root of 37 is 3.33222 and the cube root of 512 is 8. Therefore, ∛(37/512) = (∛37)/(∛512) = 3.332/8 = 0.4165.

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