Chapter 7

Solve \((x+2)^{\frac{3}{4}}=27\)

Let \(g(x)=3 x+2\) and \(f(x)=\frac{x-2}{3} .\) Find each value. $$ f(g(0)) $$

Which of the following is equivalent to \((2+3 \sqrt{5})(3+3 \sqrt{5}) ?\) $$\begin{array}{llll}{\text { A. } 51} & {\text { B. } 6+9 \sqrt{5}} & {\text { C. } 6+24 \sqrt{5}} & {\text { D. } 51+15 \sqrt{5}}\end{array}$$

Open-Ended Of the equivalent expressions \(\sqrt{\frac{2}{3}}, \frac{\sqrt{2}}{\sqrt{3}},\) and \(\frac{\sqrt{6}}{3},\) which do you prefer to use for finding a decimal approximation with a calculator? Justify your reasoning.

Tell whether each equation is true for all, some, or no values of the variable. Explain your answers. $$ \sqrt[3]{x^{8}}=x^{2} $$

Simplify each number. $$-(-27)^{-\frac{4}{3}}$$

Rewrite each function to make it easy to graph using transformations of its parent function. Describe the graph. \(y=\sqrt{\frac{x}{9}+1}+5\)

Geometry Write a function that gives the length of the hypotenuse of an isosceles right triangle with side length \(s\) . Evaluate the inverse of the function to find the side length of an isosceles right triangle with a hypotenuse of 6 in.

Solve \(\sqrt{2 x+1}-\sqrt[4]{x+11}=0\)

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

Simplify each expression. Rationalize all denominators. Assume that all variables are positive. $$ \sqrt{\sqrt{16 x^{4} y^{4}}} $$

Is the product \((1-\sqrt[3]{8})(1+\sqrt[3]{8})\) a rational number? Explain.

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

Archaeology The ratio \(R\) of radioactive carbon to nonradioactive carbon left in a sample of an organism that died \(T\) years ago can be approximated by the formula \(R=A(2.7)-\frac{T}{6013}\) . Here \(A\) is the ratio of radioactive carbon to nonradioactive carbon in the living organism. What percent of \(A\) is left after 2000 years? After 4000 years? After 8000 years?

Solve \(5 \sqrt{x}+7=8\)

Geometry You toss a pebble into a pool of water and watch the circular ripples radiate outward. You find that the function \(r(x)=12.5 x\) describe the radius \(r\) in inches of a circle \(x\) seconds after it was formed. The function \(A(x)=\pi x^{2}\) describes the area \(A\) of a circle with radius \(x .\) a. Find \((A \circ r)(x)\) when \(x=2 .\) Interpret your answer. b. Find the area of a circle 4 seconds after it was formed.

Simplify each expression. Rationalize all denominators. Assume that all variables are positive. $$ \sqrt[3]{\sqrt{64 x^{6} y^{12}}} $$

What is the value of \(\frac{2}{5+2 \sqrt{2}}-\frac{3}{5-2 \sqrt{2}} ?\) Show your work.

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

Multiple Choice The expression 0.036\(m^{\frac{3}{4}}\) is used in the study of fluids. Which best represents the value of the expression for \(m=46 \times 10^{4} ?\) \(\begin{array}{lllll}{\text { A } 636} & {\text { B } 1460} & {\text { C } 1660} & {\text { D } 16,600}\end{array}\)

Multiple Choice The expression 0.036\(m^{\frac{3}{4}}\) is used in the study of fluids. Which best represents the value of the expression for \(m=46 \times 10^{4} ?\) A 636 B 1460 C 1660 D \(16,600\)

The size of a television screen is the length of the screen's diagonal \(d\) in inches. The equation \(d=\sqrt{2 A}\) models the length of a diagonal of a television screen with area \(A .\) a. Graph the equation on your calculator. b. Suppose you want to buy a new television that has twice the area of your old television. Your old television has an area of 100 in. \(2 .\) What size screen should you buy?

Find the inverse of each function. Is the inverse a function? $$ f(x)=\frac{1}{5} x^{3} $$

Solve \(-\sqrt[3]{x}+3=0\)

For each pair of functions, find \(f(g(x))\) and \(g(f(x))\) $$ f(x)=3 x, g(x)=x^{2} $$

Simplify each expression. Rationalize all denominators. Assume that all variables are positive. $$ \sqrt[3]{3} \cdot \sqrt[3]{18} $$

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

Explain the effect that \(a\) has on the graph of \(y=a \sqrt{x}\)

Find the inverse of each function. Is the inverse a function? $$ f(x)=\sqrt[3]{x-5} $$

Solve \(\sqrt{x+2}=x\)

For each pair of functions, find \(f(g(x))\) and \(g(f(x))\) $$ f(x)=x+3, g(x)=x-5 $$

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

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

Physics In the expression \(P V^{\frac{7}{5}}, P\) represents the pressure and \(V\) represents the volume of a sample of a gas. Evaluate the expression for \(P=6\) and \(V=32\) .

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. \(y=-\sqrt{2 x+8}\)

Find the inverse of each function. Is the inverse a function? $$ f(x)=\frac{\sqrt[3]{x}}{3} $$

Simplify each expression. \(64^{\frac{2}{3}}\)

For each pair of functions, find \(f(g(x))\) and \(g(f(x))\) $$ f(x)=3 x^{2}+2, g(x)=2 x $$

Simplify each expression. Rationalize all denominators. Assume that all variables are positive. $$ \frac{\sqrt{32}}{\sqrt{2}} $$

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

Simplify each expression. Assume that all variables are positive. $$x^{\frac{2}{7}} \cdot x^{\frac{3}{14}}$$

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. \(y=-\sqrt{2(4 x-3)}\)

Find the inverse of each function. Is the inverse a function? $$ f(x)=(x-2)^{3} $$

Simplify each expression. \(25^{1.5}\)

For each pair of functions, find \(f(g(x))\) and \(g(f(x))\) $$ f(x)=\frac{x-3}{2}, g(x)=2 x-3 $$

Simplify each expression. Rationalize all denominators. Assume that all variables are positive. $$ \frac{\sqrt{62}}{\sqrt{6}} $$

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

Simplify each expression. Assume that all variables are positive. $$y^{\frac{1}{2}} \cdot y^{\frac{3}{10}}$$

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. \(y=\sqrt{3 x-5}+6\)

Find the inverse of each function. Is the inverse a function? $$ f(x)=\sqrt[4]{x} $$