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Cube Root of 169

The value of the cube root of 169 rounded to 5 decimal places is 5.52877. It is the real solution of the equation x³ = 169. The cube root of 169 is expressed as ∛169 in the radical form and as (169)^⅓ or (169)^0.33 in the exponent form. The prime factorization of 169 is 13 × 13, hence, the cube root of 169 in its lowest radical form is expressed as ∛169.

Cube root of 169: 5.528774814
Cube root of 169 in Exponential Form: (169)^⅓
Cube root of 169 in Radical Form: ∛169

Cube Root of 169

1.	What is the Cube Root of 169?
2.	How to Calculate the Cube Root of 169?
3.	Is the Cube Root of 169 Irrational?
4.	FAQs on Cube Root of 169

What is the Cube Root of 169?

The cube root of 169 is the number which when multiplied by itself three times gives the product as 169. Since 169 can be expressed as 13 × 13. Therefore, the cube root of 169 = ∛(13 × 13) = 5.5288.

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How to Calculate the Value of the Cube Root of 169?

Cube Root of 169 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 = 169
Let us assume x as 5
[∵ 5³ = 125 and 125 is the nearest perfect cube that is less than 169]
⇒ x = 5
Therefore,
∛169 = 5 (5³ + 2 × 169)/(2 × 5³ + 169)) = 5.53
⇒ ∛169 ≈ 5.53
Therefore, the cube root of 169 is 5.53 approximately.

Is the Cube Root of 169 Irrational?

Yes, because ∛169 = ∛(13 × 13) and it cannot be expressed in the form of p/q where q ≠ 0. Therefore, the value of the cube root of 169 is an irrational number.

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Cube Root of 169 Solved Examples

Example 1: Given the volume of a cube is 169 in³. Find the length of the side of the cube.

Solution:

Volume of the Cube = 169 in³ = a³
⇒ a³ = 169
Cube rooting on both sides,
⇒ a = ∛169 in
Since the cube root of 169 is 5.53, therefore, the length of the side of the cube is 5.53 in.
Example 2: What is the value of ∛169 + ∛(-169)?

Solution:

The cube root of -169 is equal to the negative of the cube root of 169.
i.e. ∛-169 = -∛169

Therefore, ∛169 + ∛(-169) = ∛169 - ∛169 = 0
Example 3: Find the real root of the equation x³ − 169 = 0.

Solution:

x³ − 169 = 0 i.e. x³ = 169
Solving for x gives us,
x = ∛169, x = ∛169 × (-1 + √3i))/2 and x = ∛169 × (-1 - √3i))/2
where i is called the imaginary unit and is equal to √-1.
Ignoring imaginary roots,
x = ∛169
Therefore, the real root of the equation x³ − 169 = 0 is for x = ∛169 = 5.5288.

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FAQs on Cube Root of 169

What is the Value of the Cube Root of 169?

We can express 169 as 13 × 13 i.e. ∛169 = ∛(13 × 13) = 5.52877. Therefore, the value of the cube root of 169 is 5.52877.

What is the Value of 2 Plus 3 Cube Root 169?

The value of ∛169 is 5.529. So, 2 + 3 × ∛169 = 2 + 3 × 5.529 = 18.587. Hence, the value of 2 plus 3 cube root 169 is 18.587.

What is the Cube of the Cube Root of 169?

The cube of the cube root of 169 is the number 169 itself i.e. (∛169)³ = (169^1/3)³ = 169.

If the Cube Root of 169 is 5.53, Find the Value of ∛0.169.

Let us represent ∛0.169 in p/q form i.e. ∛(169/1000) = 5.53/10 = 0.55. Hence, the value of ∛0.169 = 0.55.

Why is the Value of the Cube Root of 169 Irrational?

The value of the cube root of 169 cannot be expressed in the form of p/q where q ≠ 0. Therefore, the number ∛169 is irrational.

Is 169 a Perfect Cube?

The number 169 on prime factorization gives 13 × 13. Here, the prime factor 13 is not in the power of 3. Therefore the cube root of 169 is irrational, hence 169 is not a perfect cube.

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