Chapter 7

Simplify. Rationalize all denominators. Assume that all the variables are positive. $$ (\sqrt{3}-\sqrt{7})(\sqrt{3}+2 \sqrt{7}) $$

Rationalize the denominator of each expression. Assume that all variables are positive. $$ \frac{\sqrt{3 x y^{2}}}{\sqrt{5 x y^{3}}} $$

Simplify each number. $$(-32)^{\frac{6}{5}}$$

Find the two real-number solutions of each equation. $$ x^{2}=100 $$

Rewrite each function to make it easy to graph using transformations of its parent function. Describe the graph. \(y=\sqrt[3]{64 x+128}\)

Solve. Check for extraneous solutions. \(3 \sqrt{2 x}-3=9\)

Let \(f(x)=x^{2}\) and \(g(x)=x-3 .\) Find each value or expression. $$ (f \circ g)(0) $$

Simplify. Rationalize all denominators. Assume that all the variables are positive. $$ (2 \sqrt{5}+3 \sqrt{2})(5 \sqrt{5}-7 \sqrt{2}) $$

Rationalize the denominator of each expression. Assume that all variables are positive. $$ \frac{\sqrt{5 x^{4} y}}{\sqrt{2 x^{2} y^{3}}} $$

Simplify each number. $$(32)^{-\frac{4}{5}}$$

Find the two real-number solutions of each equation. $$ x^{4}=1 $$

For Exercises \(31-34, f(x)=10 x-10 .\) Find each value. $$ \left(f \circ f^{-1}\right)(d) $$

Rewrite each function to make it easy to graph using transformations of its parent function. Describe the graph. \(y=\sqrt{64 x-128}-3\)

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

Let \(f(x)=x^{2}\) and \(g(x)=x-3 .\) Find each value or expression. $$ (g \circ f)(3.5) $$

Simplify. Rationalize all denominators. Assume that all the variables are positive. $$ (1+\sqrt{72})(5+\sqrt{2}) $$

Physics The formula \(F=\frac{G m, m_{2}}{r^{2}}\) relates the gravitational force \(F\) between an object of mass \(m_{1}\) and an object of mass \(m_{2}\) separated by distance \(r\) . \(G\) is a constant known as the constant of gravitation. Solve the formula for \(r\) . Rationalize the denominator.

Simplify each number. $$4^{1.5}$$

Find the two real-number solutions of each equation. $$ x^{2}=0.25 $$

Rewrite each function to make it easy to graph using transformations of its parent function. Describe the graph. \(y=\sqrt[3]{27 x-54}+1\)

Find the inverse of each function. Is the inverse a function? $$ f(x)=\frac{3 x^{2}}{4} $$

Solve. Check for extraneous solutions. \(\sqrt{2 x-1}-3=0\)

Let \(f(x)=x^{2}\) and \(g(x)=x-3 .\) Find each value or expression. $$ (f \circ g)(3.5) $$

Simplify. Rationalize all denominators. Assume that all the variables are positive. $$ (2-\sqrt{98})(3+\sqrt{18}) $$

a. Simplify \(\frac{\sqrt{2}+\sqrt{3}}{\sqrt{3}}\) by multiplying the numerator and denominator by \(\sqrt{75}\) . b. Simplify the expression in (a) by multiplying by \(\sqrt{3}\) instead of \(\sqrt{75}\) . c. Explain how you would simplify \(\frac{\sqrt{2}+\sqrt{3}}{\sqrt{98}}\) .

Simplify each number. $$16^{1.5}$$

Find the two real-number solutions of each equation. $$ x^{4}=\frac{16}{81} $$

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

Find the inverse of each function. Is the inverse a function? $$ f(x)=\sqrt{2 x-1}+3 $$

Solve. Check for extraneous solutions. \((2 x+3)^{\frac{1}{2}}-7=0\)

Let \(f(x)=x^{2}\) and \(g(x)=x-3 .\) Find each value or expression. $$ (f \circ g)\left(\frac{1}{2}\right) $$

Simplify. Rationalize all denominators. Assume that all the variables are positive. $$ (\sqrt{x}+\sqrt{3})(\sqrt{x}+2 \sqrt{3}) $$

Simplify each expression. Rationalize all denominators. Assume that all variables are positive. $$ \sqrt{5} \cdot \sqrt{40} $$

Simplify each number. $$10,000^{0.75}$$

Arrange the numbers \(\sqrt[3]{-64},-\sqrt[3]{-64}, \sqrt{64},\) and \(\sqrt[6]{64}\) in order from least to greatest.

Graph. Find the domain and the range of each function. \(y=\sqrt{x}-6\)

Find the inverse of each function. Is the inverse a function? $$ f(x)=(x+1)^{2} $$

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

Let \(f(x)=x^{2}\) and \(g(x)=x-3 .\) Find each value or expression. $$ (g \circ f)\left(\frac{1}{2}\right) $$

Simplify. Rationalize all denominators. Assume that all the variables are positive. $$ (2 \sqrt{y}-3 \sqrt{2})(4 \sqrt{y}-5 \sqrt{2}) $$

Simplify each expression. Rationalize all denominators. Assume that all variables are positive. $$ \sqrt[3]{4} \cdot \sqrt[3]{80} $$

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

Boat Building Boat builders share an old rule of thumb for sailboats. The maximum speed \(K\) in knots is 1.35 times the square root of the length \(L\) in feet of the boat's waterline. a. A customer is planning to order a sailboat with a maximum speed of 8 knots. How long should the waterline be? b. How much longer would the waterline have to be to achieve a maximum speed of 10 knots?

Graph. Find the domain and the range of each function. \(y=\sqrt{x-6}\)

Solve. Check for extraneous solutions. \((2 x+3)^{\frac{3}{4}}-3=5\)

Let \(f(x)=x^{2}\) and \(g(x)=x-3 .\) Find each value or expression. $$ (f \circ g)(c) $$

Simplify. Rationalize all denominators. Assume that all the variables are positive. $$ \frac{4+\sqrt{27}}{2-3 \sqrt{27}} $$

Simplify each expression. Rationalize all denominators. Assume that all variables are positive. $$ \sqrt{x^{5} y^{5}} \cdot 3 \sqrt{2 x^{7} y^{6}} $$

Write each expression in simplest form. Assume that all variables are positive. $$\left(x^{-4}\right)^{7}$$

Simplify each radical expression. Use absolute value symbols when needed. $$ \sqrt[3]{0.125} $$