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

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

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

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

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

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

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