Solving Exponents Equation for Integer Value

Rewrite in powers of 2 and 3: LHS: $2^{19/2} \cdot 3^4 \cdot 2^4 \cdot 3^{2m} \cdot 2^{3n} = 2^{13.5 + 3n} \cdot 3^{4 + 2m}$. RHS: $3^n \cdot 2^{4m} \cdot (2^6)^{1/4} = 3^n \cdot 2^{4m} \cdot 2^{1.5} = 2^{4m + 1.5} \cdot 3^n$.

Equating exponents of 3: $4 + 2m = n$

Equating exponents of 2: $13.5 + 3n = 4m + 1.5 \implies 4m - 3n = 12$.

Substitute $n = 4 + 2m$ into second equation: $4m - 3(4 + 2m) = 12 \implies -2m - 12 = 12 \implies -2m = 24 \implies m = -12$.