Question

find the cube root by the prime factorisation on method = 512​

Answer 1

Let us understand it step by step.

Step 1: Find the prime factors of 512

512 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2

Step 2: Pair the factors of 512 in a group of three, such that they form cubes.

512 = (2 × 2 × 2) × (2 × 2 × 2) × (2 × 2 × 2)

512 = 23 × 23 × 23

Using the law of exponent, we get;

512 = 29 [am.an = (a)m+n]

Or

512 = (23)^3 [(am)n = amn]

512 = 8^3

Step 3: Now, we will apply cube root on both the sides to take out the factor (in cubes) as a single term.

3√512 = 3√(83)

So, here the cube root is eliminated by the cube of 8.

Hence, 3√512 = 8