# Find n is the equation 52 × 54 × 56 × … × 52n = (0.008)-30.

am× an = am + n

LHS = 52 × 54 × 56 × … × 52n

= 52 + 4 + 6 + …2n

= 52(1 + 2 + 3 + …n)

{1 + 2 + 3 + …n = }

= = 5n(n + 1)

RHS = (0.008)-30 = (8× 10-3)-30

= (23× (2×5)-3)-30

= (23× 2-3× 5-3)-30

= 590

(am)n = amn

LHS = RHS

5n(n + 1) = 590

n(n + 1) = 90

n2 + n-90 = 0

Solving for n,

n = 9 and n = -10

But, n<0 is not possible

Hence, n = 9

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