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Question:

If m and n are integers such that (√2)¹⁹ 3⁴ 4² 9^m 8ⁿ = 3ⁿ 16^m (64)^1/4, then m is

Explanation:

(√2)¹⁹ 3⁴ 4² 9^m 8ⁿ = 3ⁿ 16^m (64)^1/4

∴ 2^19/2 3⁴ 2⁴ 3^2m 2³ⁿ = 3ⁿ 2^4m 2^6/4

∴ 2^19/2 3⁴ 2⁴ 3^2m 2³ⁿ = 3ⁿ 2^4m 2^3/2

∴ 2^{[(19/2) + 4 + 3n]} 3^{(4 + 2m)} = 2^{[4m + (3/2)]} 3ⁿ

Equating the exponents of 2 and 3 on both sides, we get;

Powers of 2 : (19/2) + 4 + 3n = 4m + (3/2) 

⇒ 6n + 24 = 8m ...(1)

Powers of 3 : 4 + 2m = n  ...(2)

Solving (1) and (2), we get;

m = −12.

Hence, option (b).