Evaluate: 125 a 6 3 - 125 a 6 3

We have to simplify 125 a 6 3 - 125 a 6 3 125 a 6 3 - 125 a 6 3 = 125 a 6 3 - 125 3 × a 6 3 ∵ a × b m = a m × b m = 125 a 6 1 3 - 5 3 1 3 × a 6 1 3 ∵ a m = a 1 m = 125 a 6 × 1 3 - 5 3 × 1 3 × a 6 × 1 3 ∵ a m n = a m × n = 125 a 2 - 5 1 × a 2 = 125 a 2 - 5 a 2 = 120 a 2 Hence, the required answer is 120 a 2 .

Evaluate: - 729 × - 15625 3

We have to evaluate - 729 × - 15625 3 . Now, - 729 × - 15625 3 = - 729 3 × - 15625 3 [by using a 3 × b 3 = a b 3 ] As we know, the cube root of negative number is always negative. = - 3 × 3 × 3 × 3 × 3 × 3 3 × - 5 × 5 × 5 × 5 × 5 × 5 3 [by prime factorisation] = - 3 × 3 × - 5 × 5 [Since, a × a × a 3 = a ] = - 9 × - 25 = 225 Hence, - 729 × - 15625 3 = 225

Show that: 2197 3375 3 = 2197 3 3375 3

To prove: 2197 3 3375 3 = 2197 3375 3 LHS : 2197 3 3375 3 = ( 13 × 13 × 13 ) 3 ( 15 × 15 × 15 ) 3 = 13 3 3 15 3 3 = 13 15 RHS : 2197 3375 3 = 13 × 13 × 13 15 × 15 × 15 3 = 13 3 15 3 3 = 13 3 3 15 3 3 a b 3 = a 3 b 3 = 13 15 ∴ LHS = RHS Hence proved.