The cube root of 256 is a real number whose cube is 256. Finding the cube root of a number is the inverse function of finding the cube of a number. Let x be the cube root of 256, then ∛256 = x and x3 = 256. The sign “√” is called the radical sign often used to represent the nth root of a number and “∛” is used to represent the cube root of a number.
The cube root of 256 in exponential form is written as 2561/3. Let us learn how to find the cube root of 256.
|
Cube Root of 256 |
|
|
Cube of 256 |
16,777,216 |
What is the Cube Root of 256?
The cube root of 256 is the number whose cube is 256. A perfect cube is a number which is the cube of an integer. Since we cannot find any integer whose cube is 256, therefore 256 is not a perfect cube number. Learn the cube root of 256 and also learn how to calculate the cube root of 256.
Check the cubes of numbers from 1 to 50.
How to Find the Cube Root of 256?
Let us calculate the cube root of 256. First, we shall determine it using the prime factorisation method.
Prime Factorisation Method
Following are the steps to calculate the cube root of a number using the prime factorisation method.
- Determine the prime factors of the given number.
- Make a group of three for each prime factor in the prime factorisation.
- For each group, we take that factor one time and finally multiply all the prime factors. If any factor cannot be grouped in three then it cannot be simplified further.
Let us apply these steps to find the cube root of 256.
Prime factorisation of 256 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2
The cube root of 256 = ∛256 = ∛(2 × 2 × 2 × 2 × 2 × 2 × 2 × 2) = 2 × 2 × ∛(2 × 2) = 4∛4.
Since ∛4 cannot be simplified further, and ∛4 is an irrational number the cube root of 256 is an irrational number.
Approximation Method
Also known as Halley’s method to find an approximate cube root of any number. Let us calculate the cube root of 256 by Halley’s method whose formula is given by:
Halley Formula for Cube Root:
|
|
Where a = the number whose cube root is to be determined.
x = is the guess number which is the cube root of the nearest perfect cube.
Here a = 256 and 63 = 216 < 256 we take x = 6. Let us put these values in the above formula, we get
∛256 = 6[(63 + 2 × 256)/(2 × 63 + 256)]
= 6[(216 + 512)/(2 × 216 + 256)]
= 6[(728)/(688)] ≈ 6.348, which is approximately equal to the actual cube root of 256.
Hence, the cube root of 256 using Halley’s method is 6.348 (approx)
Video Lesson on Finding Cube Roots
Related Articles
Solved Examples on Cube Root of 256
Example 1:
What is the smallest number that must be multiplied by 256 to make it a perfect cube number?
Solution:
The prime factorisation of 256 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2
To find the cube root of 256 we need to make a group of 3 for each factor. Therefore, we must multiply 256 by 2 to get a perfect cube number.
256 × 2 = 512 is a perfect cube number.
The cube root of 512 = ∛(2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2) = 2 × 2 × 2 = 8.
Example 2:
Find the length of the side of the cube whose volume is 256 cm3.
Solution:
Let x be the side of the cube.
Volume of the cube = x3 = 256 cm3
⇒ x = ∛256 = 6.34 cm (approx.)
Example 3:
What is the smallest number that must be subtracted from 256 to get a perfect cube number?
Solution:
Now, 63 = 216 < 256.
Therefore, we must subtract 40 from 256 to get a perfect cube number.
Frequently Asked Questions on Cube Root of 256
What is the cube root of 256?
The cube root of 256 is approximately equal to 6.349.
Is 256 a perfect cube number?
No, 256 is not a perfect cube number as there does not exist any such integer whose cube is 256.
Is the cube root of 256 a whole number?
No, the cube root of 256 is not a whole number, ∛256 = 6.349604208 (approx.).
Is the cube root of 256 a rational number?
No, the cube root of 256 is an irrational number.
What is the cube number of 256?
The cube of 256 is 16,777,216.
What is the smallest number that must be added to 256 to get a perfect cube number?
Since 73 = 343 > 256, we must add 87 to 256 to get a perfect cube number.
