Cube Root of 361
The value of the cube root of 361 rounded to 7 decimal places is 7.1203674. It is the real solution of the equation x^{3} = 361. The cube root of 361 is expressed as ∛361 in the radical form and as (361)^{⅓} or (361)^{0.33} in the exponent form. The prime factorization of 361 is 19 × 19, hence, the cube root of 361 in its lowest radical form is expressed as ∛361.
 Cube root of 361: 7.120367359
 Cube root of 361 in Exponential Form: (361)^{⅓}
 Cube root of 361 in Radical Form: ∛361
1.  What is the Cube Root of 361? 
2.  How to Calculate the Cube Root of 361? 
3.  Is the Cube Root of 361 Irrational? 
4.  FAQs on Cube Root of 361 
What is the Cube Root of 361?
The cube root of 361 is the number which when multiplied by itself three times gives the product as 361. Since 361 can be expressed as 19 × 19. Therefore, the cube root of 361 = ∛(19 × 19) = 7.1204.
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How to Calculate the Value of the Cube Root of 361?
Cube Root of 361 by Halley's Method
Its formula is ∛a ≈ x ((x^{3} + 2a)/(2x^{3} + a))
where,
a = number whose cube root is being calculated
x = integer guess of its cube root.
Here a = 361
Let us assume x as 7
[∵ 7^{3} = 343 and 343 is the nearest perfect cube that is less than 361]
⇒ x = 7
Therefore,
∛361 = 7 (7^{3} + 2 × 361)/(2 × 7^{3} + 361)) = 7.12
⇒ ∛361 ≈ 7.12
Therefore, the cube root of 361 is 7.12 approximately.
Is the Cube Root of 361 Irrational?
Yes, because ∛361 = ∛(19 × 19) and it cannot be expressed in the form of p/q where q ≠ 0. Therefore, the value of the cube root of 361 is an irrational number.
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Cube Root of 361 Solved Examples

Example 1: The volume of a spherical ball is 361π in^{3}. What is the radius of this ball?
Solution:
Volume of the spherical ball = 361π in^{3}
= 4/3 × π × R^{3}
⇒ R^{3} = 3/4 × 361
⇒ R = ∛(3/4 × 361) = ∛(3/4) × ∛361 = 0.90856 × 7.12037 (∵ ∛(3/4) = 0.90856 and ∛361 = 7.12037)
⇒ R = 6.46928 in^{3} 
Example 2: Given the volume of a cube is 361 in^{3}. Find the length of the side of the cube.
Solution:
Volume of the Cube = 361 in^{3} = a^{3}
⇒ a^{3} = 361
Cube rooting on both sides,
⇒ a = ∛361 in
Since the cube root of 361 is 7.12, therefore, the length of the side of the cube is 7.12 in. 
Example 3: Find the real root of the equation x^{3} − 361 = 0.
Solution:
x^{3} − 361 = 0 i.e. x^{3} = 361
Solving for x gives us,
x = ∛361, x = ∛361 × (1 + √3i))/2 and x = ∛361 × (1  √3i))/2
where i is called the imaginary unit and is equal to √1.
Ignoring imaginary roots,
x = ∛361
Therefore, the real root of the equation x^{3} − 361 = 0 is for x = ∛361 = 7.1204.
FAQs on Cube Root of 361
What is the Value of the Cube Root of 361?
We can express 361 as 19 × 19 i.e. ∛361 = ∛(19 × 19) = 7.12037. Therefore, the value of the cube root of 361 is 7.12037.
What is the Cube of the Cube Root of 361?
The cube of the cube root of 361 is the number 361 itself i.e. (∛361)^{3} = (361^{1/3})^{3} = 361.
Is 361 a Perfect Cube?
The number 361 on prime factorization gives 19 × 19. Here, the prime factor 19 is not in the power of 3. Therefore the cube root of 361 is irrational, hence 361 is not a perfect cube.
Why is the Value of the Cube Root of 361 Irrational?
The value of the cube root of 361 cannot be expressed in the form of p/q where q ≠ 0. Therefore, the number ∛361 is irrational.
What is the Cube Root of 361?
The cube root of 361 is equal to the negative of the cube root of 361. Therefore, ∛361 = (∛361) = (7.12) = 7.12.
How to Simplify the Cube Root of 361/64?
We know that the cube root of 361 is 7.12037 and the cube root of 64 is 4. Therefore, ∛(361/64) = (∛361)/(∛64) = 7.12/4 = 1.78.
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