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If [ (2^(n+1))^m * (2^(2n))ⁿ ] / [ (2^(m+1))ⁿ * (2^(2m))^m ] = 1, find the value of m/n.

1. 1
2. 2
3. 3
4. 4

Correct answer: 1

Solution

Numerator exponent: (n+1)m + 2n*n = mn + m + 2n². Denominator exponent: (m+1)n + 2m*m = mn + n + 2m². The expression = 2^[(mn + m + 2n²) - (mn + n + 2m²)] = 2^[m - n + 2n² - 2m²]. Set exponent = 0: m - n + 2(n² - m²) = 0 => (m - n) - 2(m - n)(m + n) = 0 => (m - n)[1 - 2(m + n)] = 0. So either m = n (giving m/n = 1) or m + n = 1/2 (not generally giving a clean ratio for integer-type answer). The intended answer is m = n => m/n = 1.

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