1. Simplify $\frac{(a^8 b^{12})^2}{(a^5 b^7)^3}$: Apply power rule: $(x^m)^n = x^{mn}$. Numerator: $(a^8)^2 (b^{12})^2 = a^{16} b^{24}$. Denominator: $(a^5)^3 (b^7)^3 = a^{15} b^{21}$. Divide powers: $a^{16} / a^{15} = a^{16-15} = a^1$ and $b^{24} / b^{21} = b^{24-21} = b^3$. Final: $a b^3$. 2. Simplify $\left(\frac{a w}{2}\right)^4 \cdot \left(\frac{w^4 a}{2}\right)^{-1}$: Rewrite powers: $\left(\frac{a w}{2}\right)^4 = \frac{(a w)^4}{2^4} = \frac{a^4 w^4}{16}$. Reciprocal for negative exponent: $\left(\frac{w^4 a}{2}\right)^{-1} = \frac{2}{w^4 a}$. Multiply: $\frac{a^4 w^4}{16} \times \frac{2}{a w^4} = \frac{a^4 w^4 \times 2}{16 a w^4} = \frac{2 a^4 w^4}{16 a w^4}$. Simplify numerator and denominator: cancel $a$ and $w^4$. Result: $\frac{2 a^{4-1}}{16} = \frac{2 a^3}{16} = \frac{a^3}{8}$. 3. Simplify $\left(\frac{x^3}{y^2}\right)^4 \cdot \frac{y^3}{x^4}$: Apply power: $\left(\frac{x^3}{y^2}\right)^4 = \frac{x^{12}}{y^{8}}$. Multiply: $\frac{x^{12}}{y^{8}} \cdot \frac{y^3}{x^4} = \frac{x^{12} y^3}{y^8 x^4} = x^{12-4} y^{3-8} = x^8 y^{-5}$. 4. Simplify $x^{-3} \cdot x^4 \cdot y^3 \cdot (x^4 y^5)^2$: Apply power inside parentheses: $(x^4 y^5)^2 = x^{8} y^{10}$. Multiply all: $x^{-3} x^{4} x^{8} y^{3} y^{10} = x^{-3+4+8} y^{3+10} = x^{9} y^{13}$. 5. Simplify $\left(\frac{x^3 y^2 z^4}{x^6 y^2}\right)^2 \cdot \frac{x^4 y^5}{z^{8}}$: Inside parentheses: $\frac{x^3 y^2 z^4}{x^6 y^2} = x^{3-6} y^{2-2} z^{4} = x^{-3} z^{4}$. Square: $(x^{-3} z^{4})^2 = x^{-6} z^{8}$. Multiply by $\frac{x^4 y^5}{z^8}$: $x^{-6} z^{8} \cdot x^{4} y^{5} z^{-8} = x^{-6+4} y^{5} z^{8-8} = x^{-2} y^{5}$. 6. Simplify $\left(\frac{m^{4} n^{3}}{p^{4}}\right)^4 \cdot \left(\frac{m^{3} n^{2}}{p^{3}}\right)^{-2}$: First term: $\frac{m^{16} n^{12}}{p^{16}}$. Second term reciprocal with power: $\left(\frac{m^{3} n^{2}}{p^{3}}\right)^{-2} = \left(\frac{p^{3}}{m^{3} n^{2}}\right)^{2} = \frac{p^{6}}{m^{6} n^{4}}$. Multiply: $\frac{m^{16} n^{12}}{p^{16}} \cdot \frac{p^{6}}{m^{6} n^{4}} = m^{16-6} n^{12-4} p^{6 -16} = m^{10} n^{8} p^{-10}$. Final answers: 1. $a b^{3}$ 2. $\frac{a^{3}}{8}$ 3. $x^{8} y^{-5}$ 4. $x^{9} y^{13}$ 5. $x^{-2} y^{5}$ 6. $m^{10} n^{8} p^{-10}$