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Cube Root of 217

The value of the cube root of 217 rounded to 5 decimal places is 6.00925. It is the real solution of the equation x³ = 217. The cube root of 217 is expressed as ∛217 in the radical form and as (217)^⅓ or (217)^0.33 in the exponent form. The prime factorization of 217 is 7 × 31, hence, the cube root of 217 in its lowest radical form is expressed as ∛217.

Cube root of 217: 6.009245007
Cube root of 217 in Exponential Form: (217)^⅓
Cube root of 217 in Radical Form: ∛217

Cube Root of 217

1.	What is the Cube Root of 217?
2.	How to Calculate the Cube Root of 217?
3.	Is the Cube Root of 217 Irrational?
4.	FAQs on Cube Root of 217

What is the Cube Root of 217?

The cube root of 217 is the number which when multiplied by itself three times gives the product as 217. Since 217 can be expressed as 7 × 31. Therefore, the cube root of 217 = ∛(7 × 31) = 6.0092.

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How to Calculate the Value of the Cube Root of 217?

Cube Root of 217 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 = 217
Let us assume x as 6
[∵ 6³ = 216 and 216 is the nearest perfect cube that is less than 217]
⇒ x = 6
Therefore,
∛217 = 6 (6³ + 2 × 217)/(2 × 6³ + 217)) = 6.01
⇒ ∛217 ≈ 6.01
Therefore, the cube root of 217 is 6.01 approximately.

Is the Cube Root of 217 Irrational?

Yes, because ∛217 = ∛(7 × 31) and it cannot be expressed in the form of p/q where q ≠ 0. Therefore, the value of the cube root of 217 is an irrational number.

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Cube Root of 217 Solved Examples

Example 1: Find the real root of the equation x³ − 217 = 0.

Solution:

x³ − 217 = 0 i.e. x³ = 217
Solving for x gives us,
x = ∛217, x = ∛217 × (-1 + √3i))/2 and x = ∛217 × (-1 - √3i))/2
where i is called the imaginary unit and is equal to √-1.
Ignoring imaginary roots,
x = ∛217
Therefore, the real root of the equation x³ − 217 = 0 is for x = ∛217 = 6.0092.
Example 2: Given the volume of a cube is 217 in³. Find the length of the side of the cube.

Solution:

Volume of the Cube = 217 in³ = a³
⇒ a³ = 217
Cube rooting on both sides,
⇒ a = ∛217 in
Since the cube root of 217 is 6.01, therefore, the length of the side of the cube is 6.01 in.
Example 3: The volume of a spherical ball is 217π in³. What is the radius of this ball?

Solution:

Volume of the spherical ball = 217π in³
= 4/3 × π × R³
⇒ R³ = 3/4 × 217
⇒ R = ∛(3/4 × 217) = ∛(3/4) × ∛217 = 0.90856 × 6.00925 (∵ ∛(3/4) = 0.90856 and ∛217 = 6.00925)
⇒ R = 5.45976 in³

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FAQs on Cube Root of 217

What is the Value of the Cube Root of 217?

We can express 217 as 7 × 31 i.e. ∛217 = ∛(7 × 31) = 6.00925. Therefore, the value of the cube root of 217 is 6.00925.

Is 217 a Perfect Cube?

The number 217 on prime factorization gives 7 × 31. Here, the prime factor 7 is not in the power of 3. Therefore the cube root of 217 is irrational, hence 217 is not a perfect cube.

If the Cube Root of 217 is 6.01, Find the Value of ∛0.217.

Let us represent ∛0.217 in p/q form i.e. ∛(217/1000) = 6.01/10 = 0.6. Hence, the value of ∛0.217 = 0.6.

What is the Cube of the Cube Root of 217?

The cube of the cube root of 217 is the number 217 itself i.e. (∛217)³ = (217^1/3)³ = 217.

What is the Value of 12 Plus 6 Cube Root 217?

The value of ∛217 is 6.009. So, 12 + 6 × ∛217 = 12 + 6 × 6.009 = 48.054. Hence, the value of 12 plus 6 cube root 217 is 48.054.

What is the Cube Root of -217?

The cube root of -217 is equal to the negative of the cube root of 217. Therefore, ∛-217 = -(∛217) = -(6.009) = -6.009.

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