Solve 27^2*32^2/16^3*3^2=2^m/2^n

Evaluate

2 7^{2} \times \frac{3 2 ^{2}}{1 6 ^{3}} \times 3^{2} = \frac{2 ^{m}}{2 ^{n}}

Simplify

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\frac{3 ^{8}}{2 ^{2}} = \frac{2 ^{m}}{2 ^{n}}

Simplify

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\frac{3 ^{8}}{2 ^{2}} = 2^{m - n}

Swap the sides of the equation

2^{m - n} = \frac{3 ^{8}}{2 ^{2}}

Take the logarithm of both sides

lo g_{2} (2^{m - n}) = lo g_{2} (\frac{3 ^{8}}{2 ^{2}})

Evaluate the logarithm

m - n = lo g_{2} (\frac{3 ^{8}}{2 ^{2}})

Move the expression to the right-hand side and change its sign

m = lo g_{2} (\frac{3 ^{8}}{2 ^{2}}) + n

Solution

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m = 8 lo g_{2} (3) - 2 + n

Solve the equation