Chapter 7

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

Simplify each number. $$243^{1.2}$$

Simplify each radical expression. Use absolute value symbols when needed. $$ \sqrt[2 n]{x^{4 n}} $$

a. The graph of \(y=\sqrt{x}\) is translated five units to the right and two units down. Write an equation of the translated function. b. The translated graph from part (a) is again translated, this time four units left and three units down. Write an equation of the translated function.

For each function \(f,\) find \(f^{-1},\) the domain and range of \(f\) and \(f^{-1},\) and determine whether \(f^{-1}\) is a function. $$ f(x)=(7-x)^{2} $$

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

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

Error Analysis A student used the steps shown below to simplify an expression. Find the student's error and explain why the step is incorrect. $$ \begin{aligned} \frac{1}{(1-\sqrt{2})^{2}} &=(1-\sqrt{2})^{-2} \\\ &=1^{-2}-(\sqrt{2})^{-2} \\\ &=\frac{1}{1^{2}}-\frac{1}{\left(\sqrt{2}^{2}\right)}=\frac{1}{1}-\frac{1}{2}=\frac{1}{2} \end{aligned} $$

Simplify each expression. Rationalize all denominators. Assume that all variables are positive. $$ \frac{\sqrt{3}-\sqrt{2}}{\sqrt{8}} $$

Simplify each number. $$64^{3.5}$$

Simplify each radical expression. Use absolute value symbols when needed. $$ \sqrt[2 n]{x^{6 n}} $$

a. Graph \(y=\sqrt{x-2}+1\) and \(y=-\sqrt{x-2}+1\) b. Find the domain and the range of each function.

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

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

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

In the expression \(\sqrt[n]{x^{m}}, m\) and \(n\) are positive integers and \(x\) is a real number. The expression can be simplified. a. If \(x>0,\) what are the possible values for \(m\) and \(n\) ? b. If \(x<0,\) what are the possible values for \(m\) and \(n\) ? c. If \(x<0\) and an absolute value symbol is needed in the simplified expression, what are the possible values of \(m\) and \(n ?\)

Satellites The circular velocity \(v,\) in miles per hour, of a satellite orbiting Earth is given by the formula \(v=\sqrt{\frac{1.24 \times 10^{12}}{r}, \text { where } r \text { is the distance }}\) in miles from the satellite to the center of Earth. How much greater is the velocity of a satellite orbiting at an altitude of 100 \(\mathrm{mi}\) than one orbiting at an altitude of 200 \(\mathrm{mi} ?\) (The radius of Earth is 3950 \(\mathrm{mi}\) .)

Simplify each number. $$100^{4.5}$$

Rewrite each function to make it easy to graph using transformations of its parent function. Describe the graph. \(y=\sqrt{25 x-100}-1\)

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

Solve. Check for extraneous solutions. \(\sqrt{\sqrt{x+25}}=\sqrt{x+5}\)

Let \(f(x)=3 x^{2}+2 x-8\) and \(g(x)=x+2 .\) Perform each function operation and then find the domain. $$ \frac{5 f(x)}{g(x)} $$

Which expression does NOT simplify to one term? $$\begin{array}{ll}{\text { A. }-4 \sqrt{8}+\sqrt{18}} & {\text { B. } \sqrt{27}-\sqrt{8}} \\ {\text { C. } \sqrt{32}+3 \sqrt{8}} & {\text { D. } \sqrt{12}-\sqrt{75}}\end{array}$$

Geometry A rectangular shelf is \(\sqrt{440} \mathrm{cm}\) by \(\sqrt{20} \mathrm{cm} .\) Find its area.

Critical Thinking For what positive integers \(n\) is each of the statements true?. a. If \(x^{n}=b,\) then \(x\) is an \(n\) th root of \(b\) b. If \(x^{n}=b,\) then \(x=\sqrt[n]{b}\)

Simplify each number. $$32^{-0.4}$$

Rewrite each function to make it easy to graph using transformations of its parent function. Describe the graph. \(y=\sqrt{36 x+108}+4\)

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

Devise a plan to find the value of \(x .\) \(x=\sqrt{2+\sqrt{2+\sqrt{2+\ldots}}}\)

Writing Evaluate \((g \circ f)(3),\) when \(f(x)=2 x\) and \(g(x)=x+1 .\) Explain what you do first and why.

What is an expression for \(\sqrt{20}-\sqrt{80}+\sqrt{125} ?\) $$\begin{array}{llll}{\text { F. } \sqrt{65}} & {\text { G. } 13 \sqrt{5}} & {\text { H. } 11 \sqrt{5}} & {\text { J. } 3 \sqrt{5}}\end{array}$$

Error Analysis Explain the error in this simplification of radical expressions. \(\sqrt{-2} \cdot \sqrt{-8}=\sqrt{-2(-8)}=\sqrt{16}=4\)

Writing Is 10 a first root of 10\(?\) Explain.

Simplify each number. $$64^{-0.5}$$

Rewrite each function to make it easy to graph using transformations of its parent function. Describe the graph. \(y=-\sqrt[3]{8 x-2}\)

You have solved equations containing square roots by squaring both sides. You were using the property that if \(a=b\) then \(a^{2}=b^{2}\). Show that the following statements are not true for all real numbers. a. If \(a^{2}=b^{2}\) then \(a=b\) b. If \(a

Let \(g(x)=3 x+2\) and \(f(x)=\frac{x-2}{3} .\) Find each value. $$ f(g(1)) $$

Which expression is NOT equal to 13\(?\) A. \((4+\sqrt{3})(4-\sqrt{3})\) C. \((6+\sqrt{23})(6-\sqrt{23})\) B. \((5-2 \sqrt{3})(5+2 \sqrt{3})\) D. \((7-\sqrt{6})(7+\sqrt{6})\)

Physics A freely falling object hit the ground in \(\sqrt{18 a^{5}}\) seconds. It fell \(h\) feet. Use the formula \(h=16 t^{2}\) to find \(h\) in terms of \(a .\)

Tell whether each equation is true for all, some, or no values of the variable. Explain your answers. $$ \sqrt{x^{4}}=x^{2} $$

Simplify each number. $$64^{-0.5}$$

Rewrite each function to make it easy to graph using transformations of its parent function. Describe the graph. \(y=\sqrt{\frac{x-1}{4}}-2\)

Solve \(\sqrt{4 x-23}-3=2\)

Let \(g(x)=3 x+2\) and \(f(x)=\frac{x-2}{3} .\) Find each value. $$ g(f(-4)) $$

How can you write \(\frac{1+\sqrt{3}}{5}-\sqrt{3}\) with a rationalized denominator? \(\mathbf{F}-1\) G. \(-2-3 \sqrt{3}\) H. \(4+3 \sqrt{3}\) J. \(\frac{4+3 \sqrt{3}}{11}\)

Writing Does \(\sqrt{x^{3}}=\sqrt[3]{x^{2}}\) for all, some, or no values of \(x ?\) Explain.

Tell whether each equation is true for all, some, or no values of the variable. Explain your answers. $$ \sqrt{x^{6}}=x^{3} $$

Simplify each number. $$2(16)^{\frac{3}{4}}$$

Rewrite each function to make it easy to graph using transformations of its parent function. Describe the graph. \(y=10-\sqrt[3]{\frac{x+3}{27}}\)

Critical Thinking Relation \(r\) has one element in its domain and two elements in its range. Is \(r\) a function? Is the inverse of \(r\) a function? Explain.