Square Root and Cube Root

Square root

When a factor of a number is multiplied by itself gives the original number, then the factor number is called a square root. It is denoted by the symbol √.
For example, the square of 2 is 4, then the square root of 4 is 2. [√4 = 2]
Square of 6 is 36, then the square root of 36 is 6. [√36 = 6]

Properties of the square root

1. The numbers ending with 2, 3, 7, 8 will not be a perfect square root, and their square will not be a natural number.
2. If the number ends with odd numbers of zeroes, then its square root is not a natural number.
3. The square root of an even number is even and for an odd number is odd.
4. There is no square root for a negative number.

Cube root

When a factor of a number multiplied three times by itself gives the original number, the factor is called a cube root. It is denoted by ∛.
For example, the cube of 2 is 8, then the cube root of 8 is 2. [∛8 = 2]
Cube of 5 is 125, then the cube root of 125 is 5. [∛125 = 5]

Properties of cube root

1. The cubic root of odd numbers is an odd number. E.g. ∛27 = 3, ∛125 = 5
2. The cubic root of natural even numbers is an even number. E.g. ∛8 = 2, ∛64 = 4
3. The cubic root of a negative number is also a negative integer.

Prime factorization method to find cube roots and square roots

In this method, factor the original number into the smallest prime factors. After factoring in the numbers, make pairs of common numbers and write them only once for square root. For cube roots, make triplets of such numbers. If any number is left unpaired, then write them inside the symbol.

For example, finding the square root of 256.

Let us factor 256 into smallest prime factors as 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2

There are eight twos, which means there are four pairs of 2s.

Therefore, it can be written as (2 × 2 × 2 × 2)2 = 162

This implies, the square root of 256 is 16 and is represented by √256 = 16.

Let us find the cube root of 512 by the prime factorization method.

As said, factor 512 into smallest prime numbers as 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2.

Now let us make triplets of these twos such that there are three triplets of twos, i.e. (2 × 2 × 2)3 = 83

Therefore, the cube root of 512 is 8 and is represented as ∛512 = 8.

Square roots and cube roots of common numbers