Chapter 7

Multiply and simplify. Assume that all variables are positive. $$ 4 \sqrt{2 x} \cdot 5 \sqrt{6 x y^{2}} $$

Write each expression in exponential form. $$\sqrt{(7 x)^{3}}$$

Graph each function. \(y=-\sqrt[3]{x+3}-1\)

Graph each relation and its inverse. $$ y=(2-x)^{2} $$

Solve. Check for extraneous solutions. \(\sqrt{3 x}=\sqrt{x+6}\)

Use each diagram to find \((g \circ f)(x) .\) Then evaluate \((g \circ f)(3)\) and \((g \circ f)(-2)\) $$ f(x)=x^{2} $$ $$ g(x)=|x+5| $$

Multiply each pair of conjugates. $$ (2 \sqrt{6}+8)(2 \sqrt{6}-8) $$

Write each expression in exponential form. $$ (\sqrt{7 x})^{3} $$

Multiply and simplify. Assume that all variables are positive. $$ 3 \sqrt[3]{5 y^{3}} \cdot 2 \sqrt[3]{50 y^{4}} $$

Simplify each radical expression. Use absolute value symbols when needed. $$ \sqrt{16 x^{2}} $$

Graph each function. \(y=2 \sqrt[3]{x-6}-9\)

Graph each relation and its inverse. $$ y=(3-2 x)^{2}-1 $$

Solve. Check for extraneous solutions. \((x+5)^{\frac{1}{2}}-(5-2 x)^{\frac{1}{4}}=0\)

Let \(g(x)=2 x\) and \(h(x)=x^{2}+4 .\) Evaluate each expression. $$ (h \circ g)(1) $$

Multiply each pair of conjugates. $$ (\sqrt{3}+\sqrt{5})(\sqrt{3}-\sqrt{5}) $$

Multiply and simplify. Assume that all variables are positive. $$ -\sqrt[3]{2 x^{2} y^{2}} \cdot 2 \sqrt[3]{15 x^{5} y} $$

Write each expression in exponential form. $$\sqrt[3]{a^{2}}$$

Simplify each radical expression. Use absolute value symbols when needed. $$ \sqrt{0.25 x^{6}} $$

Graph each function. \(y=\frac{1}{2} \sqrt[3]{x-1}+3\)

For each function \(f,\) find \(f^{-1}\) and the domain and range of \(f\) and \(f^{-1} .\) Determine whether \(f^{-1}\) is a function. $$ f(x)=3 x+4 $$

Solve. Check for extraneous solutions. \((7 x+6)^{\frac{1}{2}}=(9+4 x)^{\frac{1}{2}}\)

Let \(g(x)=2 x\) and \(h(x)=x^{2}+4 .\) Evaluate each expression. $$ (h \circ g)(-5) $$

Rationalize each denominator. Simplify the answer. $$ \frac{4}{1+\sqrt{3}} $$

Divide and simplify. Assume that all variables are positive. $$ \frac{\sqrt{500}}{\sqrt{5}} $$

Write each expression in exponential form. $$(\sqrt[3]{a})^{2}$$

Simplify each radical expression. Use absolute value symbols when needed. $$ \sqrt{x^{8} y^{18}} $$

A center-pivot irrigation system can water from 1 to 130 acres of crop land. The length \(\ell\) in feet of rotating pipe needed to irrigate \(A\) acres is given by the function \(\ell=117.75 \sqrt{A}\). a. Graph the equation on your calculator. Make a sketch of the graph. b. Find the lengths of pipe needed to irrigate \(40,80,\) and 130 acres.

For each function \(f,\) find \(f^{-1}\) and the domain and range of \(f\) and \(f^{-1} .\) Determine whether \(f^{-1}\) is a function. $$ f(x)=\sqrt{x-5} $$

Solve. Check for extraneous solutions. \(\sqrt{3 x+7}=x-1\)

Let \(g(x)=2 x\) and \(h(x)=x^{2}+4 .\) Evaluate each expression. $$ (h \circ g)(-2) $$

Rationalize each denominator. Simplify the answer. $$ \frac{4}{3 \sqrt{3}-2} $$

Divide and simplify. Assume that all variables are positive. $$ \frac{\sqrt{48 x^{3}}}{\sqrt{3 x y^{2}}} $$

Write each expression in exponential form. $$\sqrt[4]{c^{2}}$$

Simplify each radical expression. Use absolute value symbols when needed. $$ \sqrt{64 b^{48}} $$

Solve each square root equation by graphing. Round the answer to the nearest hundredth if necessary. If there is no solution, explain why. \(\sqrt{x-3}=12\)

For each function \(f,\) find \(f^{-1}\) and the domain and range of \(f\) and \(f^{-1} .\) Determine whether \(f^{-1}\) is a function. $$ f(x)=\sqrt{x+7} $$

Solve. Check for extraneous solutions. \(\sqrt{x+7}-x=1\)

Let \(g(x)=2 x\) and \(h(x)=x^{2}+4 .\) Evaluate each expression. $$ (g \circ h)(-2) $$

Rationalize each denominator. Simplify the answer. $$ \frac{5+\sqrt{3}}{2-\sqrt{3}} $$

Divide and simplify. Assume that all variables are positive. $$ \frac{\sqrt{56 x^{5} y^{5}}}{\sqrt{7 x y}} $$

Write each expression in exponential form. $$\sqrt[3]{(5 x y)^{6}}$$

Simplify each radical expression. Use absolute value symbols when needed. $$ \sqrt[3]{-64 a^{3}} $$

Solve each square root equation by graphing. Round the answer to the nearest hundredth if necessary. If there is no solution, explain why. \(\sqrt{2 x-3}=4\)

For each function \(f,\) find \(f^{-1}\) and the domain and range of \(f\) and \(f^{-1} .\) Determine whether \(f^{-1}\) is a function. $$ f(x)=\sqrt{-2 x+3} $$

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

Let \(g(x)=2 x\) and \(h(x)=x^{2}+4 .\) Evaluate each expression. $$ (g \circ h)(0) $$

Rationalize each denominator. Simplify the answer. $$ \frac{3+\sqrt{8}}{2-2 \sqrt{8}} $$

Divide and simplify. Assume that all variables are positive. $$ \frac{\sqrt[3]{250 x^{7} y^{3}}}{\sqrt[3]{2 x^{2} y}} $$

The optimal height \(h\) of the letters of a message printed on pavement is given by the formula \(h=\frac{0.0052 d^{227}}{e} .\) Here \(d\) is the distance of the driver from the letters and \(e\) is the height of the driver's eye above the pavement. All of the distances are in meters. Find \(h\) for the given values of \(d\) and \(e .\) $$d=100 \mathrm{m}, e=1.2 \mathrm{m}$$

Simplify each radical expression. Use absolute value symbols when needed. $$ \sqrt[3]{27 y^{6}} $$