Chapter 7

Simplify each expression. \(6^{\frac{1}{2}} \cdot 12^{\frac{1}{2}}\)

For each pair of functions, find \(f(g(x))\) and \(g(f(x))\) $$ f(x)=-x-7, g(x)=4 x $$

Critical Thinking When \(\sqrt{x^{a} y^{b}}\) is simplified, the result is \(\frac{1}{x^{c} y^{3 d}}\) , where \(c\) and \(d\) are positive integers. Express \(a\) in terms of \(c,\) and \(b\) in terms of \(d .\)

Simplify each expression. Rationalize all denominators. Assume that all variables are positive. $$ \sqrt[3]{2 x^{2}} \cdot \sqrt[3]{4 x} $$

Simplify each radical expression. \(n\) is an odd number. $$ \sqrt[n]{m^{2 n}} $$

Simplify each expression. Assume that all variables are positive. $$x^{\frac{3}{5}} \div x^{\frac{1}{10}}$$

Rewrite each function to make it easy to graph using transformations of its parent function. Describe the graph. Find the domain and the range of each function. \(y=-3-\sqrt{12 x+18}\)

Find the inverse of each function. Is the inverse a function? $$ f(x)=1.2 x^{4} $$

Simplify each expression. \(8^{\frac{1}{2}} \cdot 40^{\frac{1}{2}}\)

For each pair of functions, find \(f(g(x))\) and \(g(f(x))\) $$ f(x)=\frac{x+5}{2}, g(x)=x^{2} $$

Critical Thinking In Example 3 you saw that \(\sqrt[3]{54 x^{2} y^{3}} \cdot \sqrt[3]{5 x^{3} y^{4}}\) simplifies to 3\(x y^{2} \sqrt[3]{10 x^{2} y, \text { if you assume that all the variables are positive. Now assume that }}\) the variables represent any real numbers. What changes must be made in the answer? Explain.

Simplify each expression. Rationalize all denominators. Assume that all variables are positive. $$ \sqrt{7 x} \cdot \sqrt{14 x^{3}} $$

Simplify each radical expression. \(n\) is an odd number. $$ \sqrt[n]{m^{3 n}} $$

Simplify each expression. Assume that all variables are positive. $$y^{\frac{5}{7}} \div y^{\frac{3}{4}}$$

a. Graph \(y=\sqrt{-x}, y=\sqrt{1-x},\) and \(y=\sqrt{2-x}\) b. How does the graph of \(y=\sqrt{h-x}\) differ from the graph of \(y=\sqrt{x-h} ?\)

What is the inverse of \(y=5 x-1 ?\) $$ \begin{array}{lllll}{\text { A. } y=5 x+1} & {\text { B. } y=\frac{x+1}{5}} & {\text { C. } y=\frac{x}{5}+1} & {\text { D. } y=\frac{x}{5}-1}\end{array} $$

Simplify each expression. \(3^{\frac{1}{3}} \cdot 18^{\frac{1}{3}}\)

Open-Ended Write a function rule that approximates each value. a. The amount you save is a percent of what you earn. (You choose the percent.) b. The amount you earn depends on how many hours you work. (You choose the hourly wage.) c. Write and simplify a composite function that expresses your savings as a function of the number of hours you work. Interpret your results.

Which expression does NOT simplify to \(-10 ?\) $$ \begin{array}{ll}{\text { A. }-\sqrt[3]{1000}} & {\text { B. } \sqrt{25} \cdot \sqrt[3]{-8}} \\ {\text { C. }-\sqrt{25} \cdot \sqrt[5]{-32}} & {\text { D. } \sqrt[3]{-125} \cdot \sqrt[4]{16}}\end{array} $$

Simplify each expression. Rationalize all denominators. Assume that all variables are positive. $$ \frac{\sqrt{6 m}}{\sqrt{2 m n}} $$

Simplify each radical expression. \(n\) is an odd number. $$ \sqrt[n]{m^{4 n}} $$

Simplify each expression. Assume that all variables are positive. $$\frac{x^{\frac{2}{3}} y^{-\frac{1}{4}}}{x^{\frac{1}{2}} y^{-\frac{1}{2}}}$$

For what positive integers \(n\) are the domain and range of \(y=\sqrt[n]{x}\) the set of real numbers? Assume that \(x\) is a real number.

If \(f(x)=4 x-3,\) what is \(\left(f^{-1} \circ f\right)(10) ?\) $$ \begin{array}{llll}{\text { E. } \frac{13}{4}} & {\text { 6. } 10} & {\text { H. } 37} & {\text { 1. } \frac{481}{4}}\end{array} $$

Simplify each expression. \(81^{-0.25}\)

How can you write \(\sqrt[3]{\frac{5}{2 x y}}\) with a rationalized denominator? \(\mathrm{F} \cdot \frac{\sqrt[3]{5}}{2 \times 4}\) G. \(\frac{\sqrt[3]{20}}{2 x y}\) H. \(\frac{\sqrt[3]{20 x^{2} y^{2}}}{2 x y}\) J. \(\frac{\sqrt[3]{4 x^{2} y^{2}}}{2 x y}\)

Simplify each expression. Rationalize all denominators. Assume that all variables are positive. $$ \sqrt[3]{\frac{4}{5 x}} $$

Which equation has more than one real-number solution? $$\begin{array}{llll}{\text { A. } x^{2}=0} & {\text { B. } x^{2}=1} & {\text { C. } x^{2}=-1} & {\text { D. } x^{3}=-1}\end{array}$$

Simplify each expression. Assume that all variables are positive. $$\frac{x^{\frac{1}{2}} y^{-\frac{1}{3}}}{x^{\frac{3}{4}} y^{\frac{1}{2}}}$$

How is the graph of \(y=\sqrt{x+7}\) translated from the graph of \(y=\sqrt{x} ?\) A. shifted 7 units left B. shifted 7 units right C. shifted 7 units up D. shifted 7 units down

What is the inverse of \(y=x^{2}-3 ?\) $$ \begin{array}{ll}{\text { A. } y=\pm \sqrt{x}+3} & {\text { B. } y=\pm \sqrt{x}-3} \\ {\text { C. } y=\pm \sqrt{x+3}} & {\text { D. } y=\pm \sqrt{x-3}}\end{array} $$

Simplify each expression. \(4^{3.5}\)

Profit A craftsman makes and sells violins. The function \(C(x)=1000+700 x\) represents his cost in dollars to produce \(x\) violins. The function \(I(x)=5995 x\) represents the income in dollars from selling \(x\) violins. a. Write and simplify a function \(P(x)=I(x)-C(x) .\) a. Find \(P(30),\) the profit earned when he makes and sells 30 violins.

What is the simplified form of \(\frac{3-\sqrt{5}}{\sqrt{5}} ?\) $$ \begin{array}{ll}{\text { A. } \frac{3 \sqrt{5}-5}{5}} & {\text { B. } \frac{5 \sqrt{3}-5}{5}} \\ {\text { C. } \sqrt{3}-\sqrt{15}} & {\text { D. } \frac{14-6 \sqrt{5}}{5}}\end{array} $$

Solve each equation. $$ 2 x^{3}-16=0 $$

Which number is greatest? $$ \begin{array}{lllll}{\text { F. } \sqrt{0.5}} & {\text { G. } \sqrt[3]{0.5}} & {\text { H. } \sqrt[4]{0.5}} & {\text { J. } \sqrt[5]{0.5}}\end{array} $$

Simplify each expression. Assume that all variables are positive. $$\left(\frac{16 x^{14}}{81 y^{18}}\right)^{\frac{1}{2}}$$

How is the graph of \(y=\sqrt{x}-5\) translated from the graph of \(y=\sqrt{x} ?\) F. shifted 5 units left G. shifted 5 units right H. shifted 5 units up J. shifted 5 units down

What is the inverse of \(y=4 x^{2}+5 ?\) For what values of \(x\) is the inverse a real number?

Simplify each expression. 125\(\cdot 125^{-\frac{1}{3}}\)

Writing A salesperson earns a 3\(\%\) bonus on weekly sales over \(\$ 5000\) . $$\begin{array}{l}{g(x)=0.03 x} \\ {h(x)=x-5000}\end{array}$$ a. Explain what each function above represents. b. Which composition, \((h \circ g)(x)\) or \((g \circ h)(x),\) represents the weekly bonus? Explain.

To rationalize the denominator of \(\sqrt[3]{\frac{2}{9}},\) by what number would you multiply the numerator and denominator of the fraction? F. 2 G. 3 H. 6 J.9

Solve each equation. $$ x^{3}+1000=0 $$

Which statement is NOT true? $$ \begin{array}{ll}{\text { A. }-3=-\sqrt{9}} & {\text { B. }-3=-\sqrt{-9}} \\\ {\text { C. }-3=\sqrt[3]{-27}} & {\text { D. }-3=-\sqrt[4]{81}}\end{array} $$

Simplify each expression. Assume that all variables are positive. $$\left(\frac{81 y^{16}}{16 x^{12}}\right)^{\frac{1}{2}}$$

The graph of \(y=-\sqrt{x}\) is shifted 4 units up and 3 units right. Which equation represents the new graph? A. \(y=-\sqrt{x-4}+3\) B. \(y=-\sqrt{x-3}+4\) C. \(y=-\sqrt{x+3}+4\) D. \(y=-\sqrt{x+4}+3\)

What is the inverse of \(y=x^{2}-2 x+1 ?\) Is the inverse a function? Explain.

Simplify each expression. 32\(\cdot 256^{-\frac{1}{2}}\)

Let \(f(x)=3 x-2\) and \(g(x)=x^{2}+1 .\) Perform each function operation and use the properties of real numbers to justify each step in simplifying your answer. $$ (f+g)(x) $$

Which of the following expressions is in simplest form? $$ \begin{array}{lllll}{\text { A. } \sqrt{20 x^{3}}} & {\text { B. } \sqrt[3]{81 x}} & {\text { C. } \sqrt{\frac{6}{2}}} & {\text { D. } \frac{\sqrt{2}}{5}}\end{array} $$