What least number must be multiplied to 3456 so that the product becomes a perfect cube?

Question

What least number must be multiplied to 6912 so that the product becomes a perfect Cube?

Hint :- By prime Factorize the given number and find the number to be multiplied

The correct answer is: 3

Explanation :-6912 = 2 × 3456= 2 × 2 × 1728= 2 × 2 × 2 × 864= 2 × 2 × 2 × 2 × 432= 2 × 2 × 2 × 2 × 2 × 216= 2 × 2 × 2 × 2 × 2 × 2 ×108= 2 × 2 × 2 × 2 × 2 × 2 × 2 × 54= 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 27= 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 ×3 × 9= 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 × 3= we know   is perfect square and to make  perfect square multiply it with 3We need to multiply 6912 with 3 to make

The value of the cube root of 3456 rounded to 5 decimal places is 15.11905. It is the real solution of the equation x3 = 3456. The cube root of 3456 is expressed as ∛3456 or 12 ∛2 in the radical form and as (3456)⅓ or (3456)0.33 in the exponent form. The prime factorization of 3456 is 2 × 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 × 3, hence, the cube root of 3456 in its lowest radical form is expressed as 12 ∛2.

• Cube root of 3456: 15.119052599
• Cube root of 3456 in Exponential Form: (3456)⅓
• Cube root of 3456 in Radical Form: ∛3456 or 12 ∛2

What is the Cube Root of 3456?

The cube root of 3456 is the number which when multiplied by itself three times gives the product as 3456. Since 3456 can be expressed as 2 × 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 × 3. Therefore, the cube root of 3456 = ∛(2 × 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 × 3) = 15.1191.

☛ Check: Cube Root Calculator

How to Calculate the Value of the Cube Root of 3456?

Cube Root of 3456 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 = 3456
Let us assume x as 15
[∵ 153 = 3375 and 3375 is the nearest perfect cube that is less than 3456]
⇒ x = 15
Therefore,
∛3456 = 15 (153 + 2 × 3456)/(2 × 153 + 3456)) = 15.12
⇒ ∛3456 ≈ 15.12
Therefore, the cube root of 3456 is 15.12 approximately.

Is the Cube Root of 3456 Irrational?

Yes, because ∛3456 = ∛(2 × 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 × 3) = 12 ∛2 and it cannot be expressed in the form of p/q where q ≠ 0. Therefore, the value of the cube root of 3456 is an irrational number.

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Cube Root of 3456 Solved Examples

1. Example 1: Given the volume of a cube is 3456 in3. Find the length of the side of the cube.

   Solution:

   Volume of the Cube = 3456 in3 = a3
   ⇒ a3 = 3456
   Cube rooting on both sides,
   ⇒ a = ∛3456 in
   Since the cube root of 3456 is 15.12, therefore, the length of the side of the cube is 15.12 in.

2. Example 2: Find the real root of the equation x3 − 3456 = 0.

   Solution:

   x3 − 3456 = 0 i.e. x3 = 3456
   Solving for x gives us,
   x = ∛3456, x = ∛3456 × (-1 + √3i))/2 and x = ∛3456 × (-1 - √3i))/2
   where i is called the imaginary unit and is equal to √-1.
   Ignoring imaginary roots,
   x = ∛3456
   Therefore, the real root of the equation x3 − 3456 = 0 is for x = ∛3456 = 15.1191.

3. Example 3: What is the value of ∛3456 ÷ ∛(-3456)?

   Solution:

   The cube root of -3456 is equal to the negative of the cube root of 3456.
   ⇒ ∛-3456 = -∛3456

   Therefore,
   ⇒ ∛3456/∛(-3456) = ∛3456/(-∛3456) = -1

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FAQs on Cube Root of 3456

What is the Value of the Cube Root of 3456?

We can express 3456 as 2 × 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 × 3 i.e. ∛3456 = ∛(2 × 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 × 3) = 15.11905. Therefore, the value of the cube root of 3456 is 15.11905.

How to Simplify the Cube Root of 3456/216?

We know that the cube root of 3456 is 15.11905 and the cube root of 216 is 6. Therefore, ∛(3456/216) = (∛3456)/(∛216) = 15.119/6 = 2.5198.

Why is the Value of the Cube Root of 3456 Irrational?

The value of the cube root of 3456 cannot be expressed in the form of p/q where q ≠ 0. Therefore, the number ∛3456 is irrational.

What is the Cube Root of -3456?

The cube root of -3456 is equal to the negative of the cube root of 3456. Therefore, ∛-3456 = -(∛3456) = -(15.119) = -15.119.

What is the Cube of the Cube Root of 3456?

The cube of the cube root of 3456 is the number 3456 itself i.e. (∛3456)3 = (34561/3)3 = 3456.

What is the Value of 11 Plus 10 Cube Root 3456?

The value of ∛3456 is 15.119. So, 11 + 10 × ∛3456 = 11 + 10 × 15.119 = 162.19. Hence, the value of 11 plus 10 cube root 3456 is 162.19.

What least number should be multiplied to 3456 to make it perfect cube?

What is the perfect cube of 3456?

What least number should be added to 3456 so as to make it a perfect square?