Find the value of m for which

(5^m)² × (25)³ × 125² = 1

(5^m)² × (25)³ × 125² = 1

⇒ (5²)^m × (25)³ × 125² = 1

⇒ 5^2m × (5²)³ × (5³)² = 1

(∵ 125 = 5× 5× 5 = 5³, and 25 = 5× 5 = 5²)

⇒ (5)^2m × (5)⁶ × (5)⁶ = 1

(∵ if ‘a’ is a real no. and m, n are integers, then (a^m)ⁿ = a^mn)

⇒ 5^{2m + 3 + 6} = 25⁰ (∵ 25⁰ = 1)

⇒ 2m + 6 + 6 = 0 (∵ base is same)

⇒ 2m + 12 = 0

⇒ 2m = -12

⇒ m = -6