Question

using prime factorization method, find which of the following numbers are not perfect squares: i) 400 ii) 768 iii) 1296 iv) 8000 v) 9025

Answer 1

Answer:

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.

Answer 2

Answer:

1. 400 is a perfect square=400=2*2*2*2*5*5=2⁴*5²