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Cube Root of 1729

The value of the cube root of 1729 rounded to 4 decimal places is 12.0023. It is the real solution of the equation x³ = 1729. The cube root of 1729 is expressed as ∛1729 in the radical form and as (1729)^⅓ or (1729)^0.33 in the exponent form. The prime factorization of 1729 is 7 × 13 × 19, hence, the cube root of 1729 in its lowest radical form is expressed as ∛1729.

Cube root of 1729: 12.002314368
Cube root of 1729 in Exponential Form: (1729)^⅓
Cube root of 1729 in Radical Form: ∛1729

Cube Root of 1729

1.	What is the Cube Root of 1729?
2.	How to Calculate the Cube Root of 1729?
3.	Is the Cube Root of 1729 Irrational?
4.	FAQs on Cube Root of 1729

What is the Cube Root of 1729?

The cube root of 1729 is the number which when multiplied by itself three times gives the product as 1729. Since 1729 can be expressed as 7 × 13 × 19. Therefore, the cube root of 1729 = ∛(7 × 13 × 19) = 12.0023.

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How to Calculate the Value of the Cube Root of 1729?

Cube Root of 1729 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 = 1729
Let us assume x as 11
[∵ 11³ = 1331 and 1331 is the nearest perfect cube that is less than 1729]
⇒ x = 11
Therefore,
∛1729 = 11 (11³ + 2 × 1729)/(2 × 11³ + 1729)) = 12.0
⇒ ∛1729 ≈ 12.0
Therefore, the cube root of 1729 is 12.0 approximately.

Is the Cube Root of 1729 Irrational?

Yes, because ∛1729 = ∛(7 × 13 × 19) and it cannot be expressed in the form of p/q where q ≠ 0. Therefore, the value of the cube root of 1729 is an irrational number.

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Cube Root of 1729 Solved Examples

Example 1: Given the volume of a cube is 1729 in³. Find the length of the side of the cube.

Solution:

Volume of the Cube = 1729 in³ = a³
⇒ a³ = 1729
Cube rooting on both sides,
⇒ a = ∛1729 in
Since the cube root of 1729 is 12.0, therefore, the length of the side of the cube is 12.0 in.
Example 2: The volume of a spherical ball is 1729π in³. What is the radius of this ball?

Solution:

Volume of the spherical ball = 1729π in³
= 4/3 × π × R³
⇒ R³ = 3/4 × 1729
⇒ R = ∛(3/4 × 1729) = ∛(3/4) × ∛1729 = 0.90856 × 12.00231 (∵ ∛(3/4) = 0.90856 and ∛1729 = 12.00231)
⇒ R = 10.90482 in³
Example 3: Find the real root of the equation x³ − 1729 = 0.

Solution:

x³ − 1729 = 0 i.e. x³ = 1729
Solving for x gives us,
x = ∛1729, x = ∛1729 × (-1 + √3i))/2 and x = ∛1729 × (-1 - √3i))/2
where i is called the imaginary unit and is equal to √-1.
Ignoring imaginary roots,
x = ∛1729
Therefore, the real root of the equation x³ − 1729 = 0 is for x = ∛1729 = 12.0023.

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FAQs on Cube Root of 1729

What is the Value of the Cube Root of 1729?

We can express 1729 as 7 × 13 × 19 i.e. ∛1729 = ∛(7 × 13 × 19) = 12.00231. Therefore, the value of the cube root of 1729 is 12.00231.

How to Simplify the Cube Root of 1729/125?

We know that the cube root of 1729 is 12.00231 and the cube root of 125 is 5. Therefore, ∛(1729/125) = (∛1729)/(∛125) = 12.002/5 = 2.4004.

If the Cube Root of 1729 is 12.0, Find the Value of ∛1.729.

Let us represent ∛1.729 in p/q form i.e. ∛(1729/1000) = 12.0/10 = 1.2. Hence, the value of ∛1.729 = 1.2.

What is the Cube of the Cube Root of 1729?

The cube of the cube root of 1729 is the number 1729 itself i.e. (∛1729)³ = (1729^1/3)³ = 1729.

What is the Cube Root of -1729?

The cube root of -1729 is equal to the negative of the cube root of 1729. Therefore, ∛-1729 = -(∛1729) = -(12.002) = -12.002.

Is 1729 a Perfect Cube?

The number 1729 on prime factorization gives 7 × 13 × 19. Here, the prime factor 7 is not in the power of 3. Therefore the cube root of 1729 is irrational, hence 1729 is not a perfect cube.

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