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Cube Root of 19

The value of the cube root of 19 rounded to 4 decimal places is 2.6684. It is the real solution of the equation x³ = 19. The cube root of 19 is expressed as ∛19 in the radical form and as (19)^⅓ or (19)^0.33 in the exponent form. The prime factorization of 19 is 19, hence, the cube root of 19 in its lowest radical form is expressed as ∛19.

Cube root of 19: 2.668401649
Cube root of 19 in Exponential Form: (19)^⅓
Cube root of 19 in Radical Form: ∛19

1.	What is the Cube Root of 19?
2.	How to Calculate the Cube Root of 19?
3.	Is the Cube Root of 19 Irrational?
4.	FAQs on Cube Root of 19

What is the Cube Root of 19?

The cube root of 19 is the number which when multiplied by itself three times gives the product as 19. The number 19 is prime. Therefore, the cube root of 19 = ∛19 = 2.6684.

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How to Calculate the Value of the Cube Root of 19?

Cube Root of 19 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 = 19
Let us assume x as 2
[∵ 2³ = 8 and 8 is the nearest perfect cube that is less than 19]
⇒ x = 2
Therefore,
∛19 = 2 (2³ + 2 × 19)/(2 × 2³ + 19)) = 2.63
⇒ ∛19 ≈ 2.63
Therefore, the cube root of 19 is 2.63 approximately.

Is the Cube Root of 19 Irrational?

Yes, because ∛19 cannot be expressed in the form of p/q where q ≠ 0. Therefore, the value of the cube root of 19 is an irrational number.

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Cube Root of 19 Solved Examples

Example 1: Find the real root of the equation x³ − 19 = 0.

Solution:

x³ − 19 = 0 i.e. x³ = 19
Solving for x gives us,
x = ∛19, x = ∛19 × (-1 + √3i))/2 and x = ∛19 × (-1 - √3i))/2
where i is called the imaginary unit and is equal to √-1.
Ignoring imaginary roots,
x = ∛19
Therefore, the real root of the equation x³ − 19 = 0 is for x = ∛19 = 2.6684.
Example 2: The volume of a spherical ball is 19π in³. What is the radius of this ball?

Solution:

Volume of the spherical ball = 19π in³
= 4/3 × π × R³
⇒ R³ = 3/4 × 19
⇒ R = ∛(3/4 × 19) = ∛(3/4) × ∛19 = 0.90856 × 2.6684 (∵ ∛(3/4) = 0.90856 and ∛19 = 2.6684)
⇒ R = 2.4244 in³
Example 3: Given the volume of a cube is 19 in³. Find the length of the side of the cube.

Solution:

Volume of the Cube = 19 in³ = a³
⇒ a³ = 19
Cube rooting on both sides,
⇒ a = ∛19 in
Since the cube root of 19 is 2.67, therefore, the length of the side of the cube is 2.67 in.

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FAQs on Cube Root of 19

What is the Value of the Cube Root of 19?

The value of the cube root of 19 is 2.6684.

Is 19 a Perfect Cube?

The number 19 is prime. Here, the prime factor 19 is not in the power of 3 and this implies that the cube root of 19 is irrational, hence 19 is not a perfect cube.

What is the Cube Root of -19?

The cube root of -19 is equal to the negative of the cube root of 19. Therefore, ∛-19 = -(∛19) = -(2.668) = -2.668.

If the Cube Root of 19 is 2.67, Find the Value of ∛0.019.

Let us represent ∛0.019 in p/q form i.e. ∛(19/1000) = 2.67/10 = 0.27. Hence, the value of ∛0.019 = 0.27.

What is the Value of 5 Plus 19 Cube Root 19?

The value of ∛19 is 2.668. So, 5 + 19 × ∛19 = 5 + 19 × 2.668 = 55.692. Hence, the value of 5 plus 19 cube root 19 is 55.692.

What is the Cube of the Cube Root of 19?

The cube of the cube root of 19 is the number 19 itself i.e. (∛19)³ = (19^1/3)³ = 19.