Question

cuberoot of 512 by prime factorisation ​

Answer 1

Answer:

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Step-by-step explanation:

The cube root of 512 is a value which results in its original number after getting multiplied by itself, three-times. It is denoted as 3√512. The cube root of a number is basically the root of a number which is cubed. If x3 = y, then 3√y = x. Thus, it is a reverse method of finding the cube of a number.

Cube root of 512, 3√512 = 8

Say, ‘n’ is the value obtained from 3√512, then n × n × n = n3 = 512 (as per the definition of cube). Since 512 is a perfect cube, we will use here the prime factorisation method, to get the cube root easily. Learn more here, to calculate the value of 3√512.

By the use of the prime factorisation method, we can find the prime factors of the given number. Now when we take the cube root of the given number, the identical or similar factors can be paired in a group of three. Hence, we will get the cubes of prime factors. Now, on applying the cube root it gets canceled by the cubed number present within it.

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 = 83

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