Q. 54.5( 2 Votes )

# Find n is the equation 5^{2} × 5^{4} × 5^{6} × … × 5^{2n} = (0.008)^{-30}.

Answer :

a^{m}× a^{n} = a^{m + n}

LHS = 5^{2} × 5^{4} × 5^{6} × … × 5^{2n}

= 5^{2 + 4 + 6 + …2n}

= 5^{2(1 + 2 + 3 + …n)}

{1 + 2 + 3 + …n = }

=

= 5^{n(n + 1)}

RHS = (0.008)^{-30} = (8× 10^{-3})^{-30}

= (2^{3}× (2×5)^{-3})^{-30}

= (2^{3}× 2^{-3}× 5^{-3})^{-30}

= 5^{90}

∵ (a^{m})^{n} = a^{mn}

LHS = RHS

∴ 5^{n(n + 1)} = 5^{90}

∴ n(n + 1) = 90

∴n^{2} + n-90 = 0

Solving for n,

n = 9 and n = -10

But, n<0 is not possible

Hence, n = 9

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