Indices problem 2 by Dhaval Furia

Algebra Level pending

If m m and n n are integers such that ( 2 ) 18 × 3 4 × 4 2 × 9 m × 8 n = 3 n × 1 6 m × 64 4 (\sqrt{2})^{18} \times 3^{4} \times 4^{2} \times 9^{m} \times 8^{n} = 3^{n} \times 16^{m} \times \sqrt[4]{64} , find m m .

− 16 -16 − 20 -20 − 24 -24 − 12 -12

Dhaval Furia by Dhaval Furia , 26, India

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1 solution

The given equation can be reduced to

2 3 n + 13.5 3 2 m + 4 = 2 4 m + 1 3 n 2^{3n+13.5}3^{2m+4}=2^{4m+1}3^n .

So, 3 n + 13.5 = 4 m + 1 3n+13.5=4m+1

2 m + 4 = n 2m+4=n

Solving we get m = − 12.25 m=-12.25 .

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