CBSE Class 8 Mathematics Square and Square Roots Notes Set 17

Perfect Squares

A perfect square is the square of a natural number (1, 4, 9, 16, ...). The unit digit of a square is 0, 1, 4, 5, 6 or 9 only; a number ending in 2, 3, 7 or 8 is never a perfect square. A number ending in 1 or 9 has a square ending in 1; ending in 4 or 6 gives 6; ending in 5 gives 5.
The square of an even number is even and the square of an odd number is odd. The square of a number ending in n zeros ends in 2n zeros. Between n^2 and (n + 1)^2 there are 2n non-square numbers.

Patterns

The sum of the first n odd numbers is n^2. (a5)^2 = a(a + 1) followed by 25, e.g. 75^2 = 5625. n^2 = (n^2 - 1)/2 + (n^2 + 1)/2 for odd n. Pythagorean triplets: 2m, m^2 - 1, m^2 + 1.
11^2 = 121, 101^2 = 10201, 1001^2 = 1002001 and so on.

Finding Square Roots

Prime factorisation: write the number as a product of primes, pair identical primes and take one from each pair.
Repeated subtraction: subtract 1, 3, 5, ... in turn; the number of steps to reach zero is the root.
Division method: pair the digits from the right (and from the decimal point in both directions), then find the root digit by digit. A number with n digits has a root with n/2 digits if n is even and (n + 1)/2 digits if n is odd.

Worked example: 5776 = 2^4 x 19^2, so the square root is 2^2 x 19 = 76. To make 252 = 2^2 x 3^2 x 7 a perfect square multiply by 7 to get 1764 = 42^2.