Using the prime factorization method, find which of the following numbers are
not perfect squares.
(i) 768
(ii) 1296
Answers
The perfect squares are : 400, 1296, 9025
Step-by-step explanation:
For a number to be a perfect square it's prime factors should be to a power of 2, or any even number.
We will write the prime factors of each of the numbers as follows :
400 = 2 × 2 × 2 × 2× 5 × 5 = 2⁴ × 5²
The prime factors have even powers hence 400 is a perfect square.
768 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 3
= 2^8 × 3
One of the prime factors (3) has an odd power hence 768 is not a perfect square.
1296 = 2 × 2 × 2 × 2 × 3 × 3 × 3 × 3
= 2⁴ × 3⁴
Since the powers of the prime factors are even then 1296 is a perfect square.
8000 = 2 × 2 × 2 × 2 × 2 × 2 × 5 × 5 × 5 = 2^6 × 5³
Since one of its prime factors has an odd power then 8000 is not a perfect square.
9025 = 5 × 5 × 19 × 19 = 5² × 19²
Since both prime factors have an even power then 9025 is a perfect square.