EXPLANATION:
Given;
We are given the expression shown below;
[tex](9^{-2})(3^{11})(\sqrt[3]{27})[/tex]Required;
We are required to express this as a single power.
Step-by-step solution;
To do this we would begin by applying some basic rules of exponents;
[tex]\begin{gathered} If: \\ a^b\times a^c \\ Then: \\ a^^{b+c} \end{gathered}[/tex]Applying it to the first two parts of the expression, we will have;
[tex]\begin{gathered} (9^{-2})=(3^2)^{-2} \\ \end{gathered}[/tex][tex](3^2)^{-2}(3^{11})=3^{-4}\times3^{11}[/tex][tex]3^{-4}\times3^{11}=3^{(-4+11)}[/tex][tex]=3^9[/tex]We will also simplify the right side of the expression as follows;
[tex]\begin{gathered} If: \\ \sqrt[a]{b^c} \\ Then: \\ b^{\frac{c}{a}} \end{gathered}[/tex][tex]\sqrt[3]{27}=27^{\frac{1}{3}}[/tex][tex]27^{\frac{1}{3}}=(3^3)^{\frac{1}{3}}[/tex][tex]\begin{gathered} If: \\ (a^b)^c \\ Then: \\ a^{b\times c} \end{gathered}[/tex]Therefore;
[tex](3^3)^{\frac{1}{3}}=3^{3\times\frac{1}{3}}[/tex][tex]3^{3\times\frac{1}{3}}=3^1[/tex]We will now combine all parts of the expression and we'll have;
[tex](9^{-2})(3^{11})(\sqrt[3]{27})[/tex][tex]=3^9\times3^1[/tex][tex]=3^{9+1}[/tex][tex]3^{10}[/tex]Also we are given the expression:
[tex]3^y5^y[/tex][tex]\begin{gathered} If: \\ a^b\times c^b \\ Then: \\ (a\times c)^b \end{gathered}[/tex]Therefore;
[tex]3^y5^y=(3\times5)^y[/tex][tex]15^y[/tex]Therefore;
ANSWER:
[tex]\begin{gathered} (a) \\ 3^{10} \end{gathered}[/tex][tex]\begin{gathered} (b) \\ 15^y \end{gathered}[/tex]