How Can We Find the Cube Root of 256?
Calculating the cube root of 256 involves certain steps. First, we'll use the prime factorisation method.
Using the Prime Factorisation Method
The prime factorisation method involves the following steps:
- Find the prime factors of the number.
- Group each prime factor in the prime factorisation into sets of three.
- Take each factor from each group once, and multiply all the prime factors. If any factor cannot be grouped into three, it can't be simplified further.
Let's apply these steps to find the cube root of 256.
Prime factorisation of 256 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2
So, the cube root of 256 = ∛256 = ∛(2 × 2 × 2 × 2 × 2 × 2 × 2 × 2) = 2 × 2 × ∛(2 × 2) = 4∛4.
Since ∛4 cannot be further simplified and is an irrational number, the cube root of 256 is an irrational number .
Using the Approximation Method
Also known as Halley’s method, this is another way to find an approximate cube root of any number. Let's calculate the cube root of 256 using Halley’s method. The formula is:
Halley Formula for Cube Root:
In this formula, 'a' represents the number whose cube root we want to determine.
'x' is a guess number which is the cube root of the closest perfect cube.
Here, 'a' = 256 and 6 3 = 216 < 256, so we take x = 6. Inserting these values into the formula, we get:
∛256 = 6[(6 3 + 2 × 256)/(2 × 6 3 + 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 approximately 6.348.