Simplify the expression using only positive exponents in the result:
(a)
[tex]\begin{gathered} (\frac{16x^4y^3}{4x^7y})^5 \\ (\frac{16^{}}{4^{}})^5.\frac{x^{4\times5}}{x^{7\times5}}\text{.}\frac{y^{3\times5}}{y^5} \\ 4^5.\frac{x^{20}}{x^{35}}.\frac{y^{15}}{y^5} \\ 1024.x^{20-35}.y^{15-5} \\ 1024x^{-15}y^{10} \\ \frac{1024y^{10}}{x^{15}} \end{gathered}[/tex]
(b)
[tex]\begin{gathered} \frac{p^{-4}(p^{-2})^{-5}}{p^2} \\ \frac{p^{-4}.p^{-2}^{\times-5}}{p^2} \\ \frac{p^{-4}.p^{10}}{p^2} \\ =p^{-4+10}.p^{-2} \\ =p^{6-2} \\ =p^4 \end{gathered}[/tex]
(c)
[tex]\begin{gathered} (3ab^3)(-5b^2c^4)(a^2c) \\ 3.(-5)a\mathrm{}a^2.\text{b}^3\text{.b}^2\text{.c}^4\text{.c} \\ -15a^{1+2}.b^{3+2}.c^{4+1} \\ -15a^3b^5c^5 \end{gathered}[/tex]
(d)
[tex]\begin{gathered} (\frac{5x^{-1}y^3}{10x^2y^5})^{-2} \\ =(\frac{5}{10}.x^{-1}.x^{-2}.y^3.y^{-5})^{-2} \\ =(\frac{1}{2}.x^{-1-2}.y^{3-5})^{-2} \\ =(\frac{1}{2}x^{-3}y^{-2})^{-2} \\ =\frac{1}{4}x^{-3(-2)}y^{-2(-2)} \\ =\frac{1}{4}x^6y^4 \end{gathered}[/tex]
