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Cube Root of 999

The value of the cube root of 999 rounded to 5 decimal places is 9.99667. It is the real solution of the equation x³ = 999. The cube root of 999 is expressed as ∛999 or 3 ∛37 in the radical form and as (999)^⅓ or (999)^0.33 in the exponent form. The prime factorization of 999 is 3 × 3 × 3 × 37, hence, the cube root of 999 in its lowest radical form is expressed as 3 ∛37.

Cube root of 999: 9.996665555
Cube root of 999 in Exponential Form: (999)^⅓
Cube root of 999 in Radical Form: ∛999 or 3 ∛37

Cube Root of 999

1.	What is the Cube Root of 999?
2.	How to Calculate the Cube Root of 999?
3.	Is the Cube Root of 999 Irrational?
4.	FAQs on Cube Root of 999

What is the Cube Root of 999?

The cube root of 999 is the number which when multiplied by itself three times gives the product as 999. Since 999 can be expressed as 3 × 3 × 3 × 37. Therefore, the cube root of 999 = ∛(3 × 3 × 3 × 37) = 9.9967.

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How to Calculate the Value of the Cube Root of 999?

Cube Root of 999 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 = 999
Let us assume x as 9
[∵ 9³ = 729 and 729 is the nearest perfect cube that is less than 999]
⇒ x = 9
Therefore,
∛999 = 9 (9³ + 2 × 999)/(2 × 9³ + 999)) = 9.99
⇒ ∛999 ≈ 9.99
Therefore, the cube root of 999 is 9.99 approximately.

Is the Cube Root of 999 Irrational?

Yes, because ∛999 = ∛(3 × 3 × 3 × 37) = 3 ∛37 and it cannot be expressed in the form of p/q where q ≠ 0. Therefore, the value of the cube root of 999 is an irrational number.

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Cube Root of 999 Solved Examples

Example 1: Given the volume of a cube is 999 in³. Find the length of the side of the cube.

Solution:

Volume of the Cube = 999 in³ = a³
⇒ a³ = 999
Cube rooting on both sides,
⇒ a = ∛999 in
Since the cube root of 999 is 10.0, therefore, the length of the side of the cube is 10.0 in.
Example 2: Find the real root of the equation x³ − 999 = 0.

Solution:

x³ − 999 = 0 i.e. x³ = 999
Solving for x gives us,
x = ∛999, x = ∛999 × (-1 + √3i))/2 and x = ∛999 × (-1 - √3i))/2
where i is called the imaginary unit and is equal to √-1.
Ignoring imaginary roots,
x = ∛999
Therefore, the real root of the equation x³ − 999 = 0 is for x = ∛999 = 9.9967.
Example 3: What is the value of ∛999 ÷ ∛(-999)?

Solution:

The cube root of -999 is equal to the negative of the cube root of 999.
⇒ ∛-999 = -∛999

Therefore,
⇒ ∛999/∛(-999) = ∛999/(-∛999) = -1

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FAQs on Cube Root of 999

What is the Value of the Cube Root of 999?

We can express 999 as 3 × 3 × 3 × 37 i.e. ∛999 = ∛(3 × 3 × 3 × 37) = 9.99667. Therefore, the value of the cube root of 999 is 9.99667.

Why is the Value of the Cube Root of 999 Irrational?

The value of the cube root of 999 cannot be expressed in the form of p/q where q ≠ 0. Therefore, the number ∛999 is irrational.

What is the Cube Root of -999?

The cube root of -999 is equal to the negative of the cube root of 999. Therefore, ∛-999 = -(∛999) = -(9.997) = -9.997.

What is the Cube of the Cube Root of 999?

The cube of the cube root of 999 is the number 999 itself i.e. (∛999)³ = (999^1/3)³ = 999.

How to Simplify the Cube Root of 999/8?

We know that the cube root of 999 is 9.99667 and the cube root of 8 is 2. Therefore, ∛(999/8) = (∛999)/(∛8) = 9.997/2 = 4.9985.

If the Cube Root of 999 is 10.0, Find the Value of ∛0.999.

Let us represent ∛0.999 in p/q form i.e. ∛(999/1000) = 10.0/10 = 1.0. Hence, the value of ∛0.999 = 1.0.

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