Simplify the expression: (2^mn 16^mn) / (8^(m-n) 2^(m+n))

Step 1: Rewrite all bases as powers of 2. We know that $16 = 2^4$ and $8 = 2^3$. Substitute these into the expression: $\frac{2^{mn} \times (2^4)^{mn}}{(2^3)^{m-n} \times 2^{m+n}}$

Step 2: Apply the exponent rule $(a^b)^c = a^{bc}$. $\frac{2^{mn} \times 2^{4 \times mn}}{2^{3 \times (m-n)} \times 2^{m+n}}$ $\frac{2^{mn} \times 2^{4mn}}{2^{3m-3n} \times 2^{m+n}}$

Step 3: Apply the exponent rule $a^b \times a^c = a^{b+c}$ to combine terms in the numerator and denominator. Numerator: $2^{mn} \times 2^{4mn} = 2^{mn + 4mn} = 2^{5mn}$ Denominator: $2^{3m-3n} \times 2^{m+n} = 2^{(3m-3n) + (m+n)} = 2^{3m-3n+m+n} = 2^{4m-2n}$ The expression now is: $\frac{2^{5mn}}{2^{4m-2n}}$

Step 4: Apply the exponent rule $\frac{a^b}{a^c} = a^{b-c}$. $2^{5mn - (4m-2n)}$

Step 5: Simplify the exponent. $2^{5mn - 4m + 2n}$

The simplified expression is: $2^{5mn - 4m + 2n}$