Chapter 7

Write each expression in simplest form. Assume that all variables are positive. $$\left(x^{\frac{1}{2}} y^{-\frac{2}{3}}\right)^{-6}$$

Simplify each radical expression. Use absolute value symbols when needed. $$ \sqrt[5]{y^{20}} $$

Graph. Find the domain and the range of each function. \(y=-\sqrt[3]{8 x}+5\)

For each function \(f,\) find \(f^{-1},\) the domain and range of \(f\) and \(f^{-1},\) and determine whether \(f^{-1}\) is a function. $$ f(x)=-\sqrt{x} $$

Let \(f(x)=2 x+5\) and \(g(x)=x^{2}-3 x+2 .\) Perform each function operation. $$ g(x)-f(x) $$

Writing Discuss the advantages and disadvantages of first simplifying \(\sqrt{72}+\sqrt{32}+\sqrt{18}\) in order to estimate its decimal value.

Simplify each expression. Rationalize all denominators. Assume that all variables are positive. $$ \frac{5 \sqrt{2}}{\sqrt{7 x}} $$

Write each expression in simplest form. Assume that all variables are positive. $$\left(x^{\frac{2}{3}} y^{-\frac{1}{6}}\right)^{-12}$$

Simplify each radical expression. Use absolute value symbols when needed. $$ \sqrt[5]{-y^{20}} $$

Graph. Find the domain and the range of each function. \(y=-2 \sqrt[3]{x-4}\)

For each function \(f,\) find \(f^{-1},\) the domain and range of \(f\) and \(f^{-1},\) and determine whether \(f^{-1}\) is a function. $$ f(x)=\sqrt{x}+3 $$

Let \(f(x)=2 x+5\) and \(g(x)=x^{2}-3 x+2 .\) Perform each function operation. $$ -2 g(x)+f(x) $$

Physics An object is moving at a speed of \((3+\sqrt{2}) \mathrm{ft} / \mathrm{s}\) . How long will it take the object to travel 20 \(\mathrm{ft}\) ?

Simplify each expression. Rationalize all denominators. Assume that all variables are positive. $$ \frac{1}{\sqrt[3]{9 x}} $$

Write each expression in simplest form. Assume that all variables are positive. $$\left(\frac{x^{\frac{1}{4}}}{y^{-\frac{3}{4}}}\right)^{12}$$

Simplify each radical expression. Use absolute value symbols when needed. $$ \sqrt[5]{k^{15}} $$

Graph. Find the domain and the range of each function. \(y=-1-\sqrt{4 x+20}\)

For each function \(f,\) find \(f^{-1},\) the domain and range of \(f\) and \(f^{-1},\) and determine whether \(f^{-1}\) is a function. $$ f(x)=\sqrt{-x+3} $$

Let \(f(x)=2 x+5\) and \(g(x)=x^{2}-3 x+2 .\) Perform each function operation. $$ f(x)-g(x)+10 $$

Simplify each expression. Rationalize all denominators. Assume that all variables are positive. $$ \frac{10}{\sqrt[3]{5 x^{2}}} $$

Write each expression in simplest form. Assume that all variables are positive. $$\left(\frac{x^{-\frac{2}{3}}}{y^{-\frac{1}{3}}}\right)^{15}$$

Simplify each radical expression. Use absolute value symbols when needed. $$ \sqrt[5]{-k^{15}} $$

Graph. Find the domain and the range of each function. \(y=4-\sqrt[3]{x+2.5}\)

For each function \(f,\) find \(f^{-1},\) the domain and range of \(f\) and \(f^{-1},\) and determine whether \(f^{-1}\) is a function. $$ f(x)=\sqrt{x+2} $$

Let \(f(x)=2 x+5\) and \(g(x)=x^{2}-3 x+2 .\) Perform each function operation. $$ 4 f(x)+2 g(x) $$

Multiple Choice The length of a rectangle is \((3+\sqrt{5}) x\) . The height is \((1+2 \sqrt{5}) y .\) Which expression best describes the area of a rectangle? A \((4+3 \sqrt{5})(x+y)\) C \((6+2 \sqrt{5}) x+(2+4 \sqrt{5}) y\) B 13\(x y\) D \((13+7 \sqrt{5}) x y\)

Simplify each expression. Rationalize all denominators. Assume that all variables are positive. $$ \frac{\sqrt[3]{14}}{\sqrt[3]{7 x^{2} y}} $$

Simplify each number. $$(-343)^{\frac{1}{3}}$$

Simplify each radical expression. Use absolute value symbols when needed. $$ \sqrt{(x+3)^{2}} $$

Graph. Find the domain and the range of each function. \(y=-3 \sqrt{x-\frac{3}{4}}+7\)

For each function \(f,\) find \(f^{-1},\) the domain and range of \(f\) and \(f^{-1},\) and determine whether \(f^{-1}\) is a function. $$ f(x)=\frac{x^{2}}{2} $$

The velocity \(v\) of an object dropped from a tall building is given by the formula \(v=\sqrt{64 d},\) where \(d\) is the distance the object has dropped. Solve the formula for \(d\)

Let \(f(x)=3 x^{2}+2 x-8\) and \(g(x)=x+2 .\) Perform each function operation and then find the domain. $$ -f(x)+4 g(x) $$

Add or subtract. $$ \frac{1}{1-\sqrt{5}}+\frac{1}{1+\sqrt{5}} $$

Simplify each expression. Rationalize all denominators. Assume that all variables are positive. $$ \frac{3 \sqrt{11 x^{3} y}}{-2 \sqrt{12 x^{4} y}} $$

Simplify each number. $$(-243)^{\frac{1}{5}}$$

Simplify each radical expression. Use absolute value symbols when needed. $$ \sqrt{(x+1)^{4}} $$

The time \(t\) in seconds for a trapeze to complete one full cycle is given by the function \(t=1.11 \sqrt{\ell}\) , where \(\ell\) is the length of the trapeze in feet. a. Graph the equation on your calculator. Make a sketch of the graph. b. How long is a full cycle if the trapeze is 15 ft. long? 30 ft. long?

For each function \(f,\) find \(f^{-1},\) the domain and range of \(f\) and \(f^{-1},\) and determine whether \(f^{-1}\) is a function. $$ f(x)=\frac{1}{x^{2}} $$

Write an equation that has two radical expressions and no real roots.

Let \(f(x)=3 x^{2}+2 x-8\) and \(g(x)=x+2 .\) Perform each function operation and then find the domain. $$ f(x)-2 g(x) $$

Add or subtract. $$ \frac{4}{\sqrt{5}-\sqrt{3}}-\frac{4}{\sqrt{5}+\sqrt{3}} $$

Simplify each expression. Rationalize all denominators. Assume that all variables are positive. $$ -2(\sqrt[3]{32}+\sqrt[3]{54}) $$

Simplify each number. $$32^{1.2}$$

Simplify each radical expression. Use absolute value symbols when needed. $$ \sqrt[2 n]{x^{2 n}} $$

a. Graph \(y=\sqrt{x-2}-2\) b. Find the domain and the range. b. At what coordinate point des the graph start? d. What is the relationship of the point at which the graph starts to the domain and the range?

For each function \(f,\) find \(f^{-1},\) the domain and range of \(f\) and \(f^{-1},\) and determine whether \(f^{-1}\) is a function. $$ f(x)=(x-4)^{2} $$

Solve. Check for extraneous solutions. \(\sqrt{x+1}+\sqrt{2 x}=\sqrt{5 x+3}\)

Let \(f(x)=3 x^{2}+2 x-8\) and \(g(x)=x+2 .\) Perform each function operation and then find the domain. $$ f(x) \cdot g(x) $$

For what values of \(a\) and \(b\) does \(\sqrt{a}+\sqrt{b}=\sqrt{a+b} ?\)