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Cube Root of 42

The value of the cube root of 42 rounded to 4 decimal places is 3.476. It is the real solution of the equation x³ = 42. The cube root of 42 is expressed as ∛42 in the radical form and as (42)^⅓ or (42)^0.33 in the exponent form. The prime factorization of 42 is 2 × 3 × 7, hence, the cube root of 42 in its lowest radical form is expressed as ∛42.

Cube root of 42: 3.476026645
Cube root of 42 in Exponential Form: (42)^⅓
Cube root of 42 in Radical Form: ∛42

Cube Root of 42

1.	What is the Cube Root of 42?
2.	How to Calculate the Cube Root of 42?
3.	Is the Cube Root of 42 Irrational?
4.	FAQs on Cube Root of 42

What is the Cube Root of 42?

The cube root of 42 is the number which when multiplied by itself three times gives the product as 42. Since 42 can be expressed as 2 × 3 × 7. Therefore, the cube root of 42 = ∛(2 × 3 × 7) = 3.476.

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How to Calculate the Value of the Cube Root of 42?

Cube Root of 42 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 = 42
Let us assume x as 3
[∵ 3³ = 27 and 27 is the nearest perfect cube that is less than 42]
⇒ x = 3
Therefore,
∛42 = 3 (3³ + 2 × 42)/(2 × 3³ + 42)) = 3.47
⇒ ∛42 ≈ 3.47
Therefore, the cube root of 42 is 3.47 approximately.

Is the Cube Root of 42 Irrational?

Yes, because ∛42 = ∛(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 42 is an irrational number.

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Cube Root of 42 Solved Examples

Example 1: Find the real root of the equation x³ − 42 = 0.

Solution:

x³ − 42 = 0 i.e. x³ = 42
Solving for x gives us,
x = ∛42, x = ∛42 × (-1 + √3i))/2 and x = ∛42 × (-1 - √3i))/2
where i is called the imaginary unit and is equal to √-1.
Ignoring imaginary roots,
x = ∛42
Therefore, the real root of the equation x³ − 42 = 0 is for x = ∛42 = 3.476.
Example 2: The volume of a spherical ball is 42π in³. What is the radius of this ball?

Solution:

Volume of the spherical ball = 42π in³
= 4/3 × π × R³
⇒ R³ = 3/4 × 42
⇒ R = ∛(3/4 × 42) = ∛(3/4) × ∛42 = 0.90856 × 3.47603 (∵ ∛(3/4) = 0.90856 and ∛42 = 3.47603)
⇒ R = 3.15818 in³
Example 3: Given the volume of a cube is 42 in³. Find the length of the side of the cube.

Solution:

Volume of the Cube = 42 in³ = a³
⇒ a³ = 42
Cube rooting on both sides,
⇒ a = ∛42 in
Since the cube root of 42 is 3.48, therefore, the length of the side of the cube is 3.48 in.

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FAQs on Cube Root of 42

What is the Value of the Cube Root of 42?

We can express 42 as 2 × 3 × 7 i.e. ∛42 = ∛(2 × 3 × 7) = 3.47603. Therefore, the value of the cube root of 42 is 3.47603.

What is the Cube of the Cube Root of 42?

The cube of the cube root of 42 is the number 42 itself i.e. (∛42)³ = (42^1/3)³ = 42.

Why is the Value of the Cube Root of 42 Irrational?

The value of the cube root of 42 cannot be expressed in the form of p/q where q ≠ 0. Therefore, the number ∛42 is irrational.

What is the Value of 16 Plus 14 Cube Root 42?

The value of ∛42 is 3.476. So, 16 + 14 × ∛42 = 16 + 14 × 3.476 = 64.664. Hence, the value of 16 plus 14 cube root 42 is 64.664.

Is 42 a Perfect Cube?

The number 42 on prime factorization gives 2 × 3 × 7. Here, the prime factor 2 is not in the power of 3. Therefore the cube root of 42 is irrational, hence 42 is not a perfect cube.

If the Cube Root of 42 is 3.48, Find the Value of ∛0.042.

Let us represent ∛0.042 in p/q form i.e. ∛(42/1000) = 3.48/10 = 0.35. Hence, the value of ∛0.042 = 0.35.

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