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Cube Root of 84

The value of the cube root of 84 rounded to 4 decimal places is 4.3795. It is the real solution of the equation x³ = 84. The cube root of 84 is expressed as ∛84 in the radical form and as (84)^⅓ or (84)^0.33 in the exponent form. The prime factorization of 84 is 2 × 2 × 3 × 7, hence, the cube root of 84 in its lowest radical form is expressed as ∛84.

Cube root of 84: 4.37951914
Cube root of 84 in Exponential Form: (84)^⅓
Cube root of 84 in Radical Form: ∛84

Cube Root of 84

1.	What is the Cube Root of 84?
2.	How to Calculate the Cube Root of 84?
3.	Is the Cube Root of 84 Irrational?
4.	FAQs on Cube Root of 84

What is the Cube Root of 84?

The cube root of 84 is the number which when multiplied by itself three times gives the product as 84. Since 84 can be expressed as 2 × 2 × 3 × 7. Therefore, the cube root of 84 = ∛(2 × 2 × 3 × 7) = 4.3795.

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How to Calculate the Value of the Cube Root of 84?

Cube Root of 84 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 = 84
Let us assume x as 4
[∵ 4³ = 64 and 64 is the nearest perfect cube that is less than 84]
⇒ x = 4
Therefore,
∛84 = 4 (4³ + 2 × 84)/(2 × 4³ + 84)) = 4.38
⇒ ∛84 ≈ 4.38
Therefore, the cube root of 84 is 4.38 approximately.

Is the Cube Root of 84 Irrational?

Yes, because ∛84 = ∛(2 × 2 × 3 × 7) and it cannot be expressed in the form of p/q where q ≠ 0. Therefore, the value of the cube root of 84 is an irrational number.

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Cube Root of 84 Solved Examples

Example 1: Find the real root of the equation x³ − 84 = 0.

Solution:

x³ − 84 = 0 i.e. x³ = 84
Solving for x gives us,
x = ∛84, x = ∛84 × (-1 + √3i))/2 and x = ∛84 × (-1 - √3i))/2
where i is called the imaginary unit and is equal to √-1.
Ignoring imaginary roots,
x = ∛84
Therefore, the real root of the equation x³ − 84 = 0 is for x = ∛84 = 4.3795.
Example 2: The volume of a spherical ball is 84π in³. What is the radius of this ball?

Solution:

Volume of the spherical ball = 84π in³
= 4/3 × π × R³
⇒ R³ = 3/4 × 84
⇒ R = ∛(3/4 × 84) = ∛(3/4) × ∛84 = 0.90856 × 4.37952 (∵ ∛(3/4) = 0.90856 and ∛84 = 4.37952)
⇒ R = 3.97906 in³
Example 3: Given the volume of a cube is 84 in³. Find the length of the side of the cube.

Solution:

Volume of the Cube = 84 in³ = a³
⇒ a³ = 84
Cube rooting on both sides,
⇒ a = ∛84 in
Since the cube root of 84 is 4.38, therefore, the length of the side of the cube is 4.38 in.

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FAQs on Cube Root of 84

What is the Value of the Cube Root of 84?

We can express 84 as 2 × 2 × 3 × 7 i.e. ∛84 = ∛(2 × 2 × 3 × 7) = 4.37952. Therefore, the value of the cube root of 84 is 4.37952.

What is the Cube of the Cube Root of 84?

The cube of the cube root of 84 is the number 84 itself i.e. (∛84)³ = (84^1/3)³ = 84.

How to Simplify the Cube Root of 84/27?

We know that the cube root of 84 is 4.37952 and the cube root of 27 is 3. Therefore, ∛(84/27) = (∛84)/(∛27) = 4.38/3 = 1.46.

Why is the Value of the Cube Root of 84 Irrational?

The value of the cube root of 84 cannot be expressed in the form of p/q where q ≠ 0. Therefore, the number ∛84 is irrational.

What is the Cube Root of -84?

The cube root of -84 is equal to the negative of the cube root of 84. Therefore, ∛-84 = -(∛84) = -(4.38) = -4.38.

What is the Value of 5 Plus 7 Cube Root 84?

The value of ∛84 is 4.38. So, 5 + 7 × ∛84 = 5 + 7 × 4.38 = 35.66. Hence, the value of 5 plus 7 cube root 84 is 35.66.

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