If (9^n × 3^2 × 3^n - (27)^n)/((3^3)^5 × 2^3) = 1/27, then find the value of n.
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if (9^n × 3^2 ×3^n × 27^n)/((3)^3)^5 × 3^2) = 1/27
then
LHS=(3^2n × 3^2 × 3^n × 3^3n)/(3^(3×5) × 3^2)
=(3^(2n+2+n+3n)/(3^(15+2))
=(3^(6n+2))/3^17
=3^(6n+2-17)
=3^(6n-15)
now,
RHS=1/27
=1/3^3
=3^(-3)
now
LHS=RHS
3^(6n-15) = 3^(-3)
then,
6n-15 = -3
6n = -3+15
n = 12/6
n = 2
☺
then
LHS=(3^2n × 3^2 × 3^n × 3^3n)/(3^(3×5) × 3^2)
=(3^(2n+2+n+3n)/(3^(15+2))
=(3^(6n+2))/3^17
=3^(6n+2-17)
=3^(6n-15)
now,
RHS=1/27
=1/3^3
=3^(-3)
now
LHS=RHS
3^(6n-15) = 3^(-3)
then,
6n-15 = -3
6n = -3+15
n = 12/6
n = 2
☺
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