Question

Can you factorize these numbers and their squares as product of prime powers?

(i) 35

(ii) 45

(iii) 72

(iv) 36

(v) 49

Anything special about the exponents in the factorization of squares?

Answer 1

(i) 35 = 5 × 7

Square of 35 = 35²

= (5 × 7)²

= 5² × 7²

(ii) 45 = 3 × 3 × 5 = 3² × 5

Square of 45 = 45²

= (3² × 5)²

= 3⁴ × 5²

(iii) 72 = 2 × 2 × 2 × 3 × 3 = 2³ × 3²

Square of 72 = 72²

= (2³ × 3²)²

= 2⁶ × 3⁴

(iv) 36 = 2 × 2 × 3 × 3 = 2² × 3²

Square of 36 = 36²

= (2² × 3²)²

= 2⁴ × 3⁴

(v) 49 = 7 × 7

Square of 49 = 49²

= (7 × 7)²

= 7² × 7²

It can be observed from the square numbers that the exponents in the factorisation of square numbers are double the exponents in the prime factorisation of the numbers.