91Ó°ÊÓ

Apply the quotient rule for exponents, if possible, and write each result using only positive exponents. Assume that all variables represent nonzero real numbers. $$ \frac{x^{3}}{y^{2}} $$

Perform the indicated operations. $$ \left[-\left(y^{4}-y^{2}+1\right)-\left(y^{4}+2 y^{2}+1\right)\right]+\left(3 y^{4}-3 y^{2}-2\right) $$

The tables give some selected ordered pairs for functions \(f\) and \(g\). $$ \begin{array}{c|c|c|c|c} x & 3 & 4 & 6 & 8 \\ \hline f(x) & 1 & 3 & 9 & 2 \end{array} $$ $$ \begin{array}{c|c|c|c|c} x & 2 & 7 & 1 & 9 \\ \hline g(x) & 3 & 6 & 9 & 12 \end{array} $$ Tables like these can be used to evaluate composite functions. For example, to evaluate \((g \circ f)(6),\) use the first table to find \(f(6)=9 .\) Then use the second table to find $$ (g \circ f)(6)=g(f(6))=g(9)=12 $$ Find each of the following. \((g \circ f)(8)\)

Perform the indicated operations. $$ [2 p-(3 p-6)]-[(5 p-(8-9 p))+4 p] $$

Find each product. $$ [((3 m-y)+z][(3 m-y)-z] $$

Apply the quotient rule for exponents, if possible, and write each result using only positive exponents. Assume that all variables represent nonzero real numbers. $$ \frac{y^{5}}{t^{3}} $$

Find each product. $$ (y+2)^{3} $$

Perform the indicated operations. $$ -\left[3 z^{2}+5 z-\left(2 z^{2}-6 z\right)\right]+\left[\left(8 z^{2}-\left[5 z-z^{2}\right]\right)+2 z^{2}\right] $$

Simplify using the power rules. Assume that all variables represent nonzero real numbers. $$ \left(x^{3}\right)^{6} $$

The tables give some selected ordered pairs for functions \(f\) and \(g\). $$ \begin{array}{c|c|c|c|c} x & 3 & 4 & 6 & 8 \\ \hline f(x) & 1 & 3 & 9 & 2 \end{array} $$ $$ \begin{array}{c|c|c|c|c} x & 2 & 7 & 1 & 9 \\ \hline g(x) & 3 & 6 & 9 & 12 \end{array} $$ Tables like these can be used to evaluate composite functions. For example, to evaluate \((g \circ f)(6),\) use the first table to find \(f(6)=9 .\) Then use the second table to find $$ (g \circ f)(6)=g(f(6))=g(9)=12 $$ Find each of the following. \((f \circ f)(4)\)