Chapter 7

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

Write each function in factored form. Check by multiplying. $$ y=81 x^{2}+36 x+4 $$

a. Reasoning Show that \(\forall f / x^{2}=\sqrt{x}\) by using the definition of fourth root. b. Reasoning Show that \(\sqrt[4]{x^{2}}=\sqrt{x}\) by rewriting \(\sqrt[4]{x^{2}}\) in exponential form.

Find each indicated root if it is a real number. $$ -\sqrt[4]{16} $$

Find each composition of functions. Simplify your answer. Let \(f(x)=\frac{1}{x} .\) Find \(f(f(f(x)))\)

Simplify each radical expression. Use absolute value symbols as needed. $$ \sqrt[7]{x^{14} y^{35}} $$

Write each function in factored form. Check by multiplying. $$ y=4 x^{3}+8 x^{2}+4 x $$

Exponents that are irrational numbers can be defined so that all the properties of rational exponents are also true for irrational exponents. Use those properties to simplify each expression. $$\left(7^{\sqrt{2}}\right)^{\sqrt{2}}$$

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

Find each indicated root if it is a real number. $$ \sqrt[4]{-16} $$

Evaluate each expression. \(_{5} \mathrm{C}_{5}\)

Find each composition of functions. Simplify your answer. Let \(f(x)=1-\frac{x}{2} .\) Find \(f(f(f(x)))\)

Simplify each radical expression. Use absolute value symbols as needed. $$ \sqrt[4]{16 x^{36} y^{96}} $$

Write each function in factored form. Check by multiplying. $$ y=12 x^{3}+14 x^{2}+2 x $$

Exponents that are irrational numbers can be defined so that all the properties of rational exponents are also true for irrational exponents. Use those properties to simplify each expression. $$\frac{3^{3+\sqrt{5}}}{3^{1+\sqrt{5}}}$$

Rationalize the denominator of each expression. Assume that all variables are positive. \(\frac{\sqrt{36 x^{3}}}{\sqrt{12 x}}\)

Find each indicated root if it is a real number. $$ \sqrt[5]{243} $$

Evaluate each expression. \(_{6} \mathrm{C}_{5}\)

Find each composition of functions. Simplify your answer. Let \(f(x)=2 x-3 .\) Find \(\frac{f(1+h)-f(1)}{h}, h \neq 0\)

Simplify each radical expression. Use absolute value symbols as needed. $$ \sqrt{0.0064 x^{40}} $$

Rewrite each equation in vertex form. $$ y=3 x^{2}-7 $$

exponents are also true for irrational exponents. Use those properties to simplify each expression. $$\frac{x^{4} \pi}{x^{2 \pi}}$$

Rationalize the denominator of each expression. Assume that all variables are positive. \(\sqrt[3]{\frac{3 x}{2 y}}\)

Find each indicated root if it is a real number. $$ -\sqrt[5]{243} $$

Evaluate each expression. \(_{7} \mathrm{C}_{1}\)

Find each composition of functions. Simplify your answer. Let \(f(x)=4 x-1 .\) Find \(\frac{f(a+h)-f(a)}{h}, h \neq 0\)

Divide. Tell whether each divisor is a factor of the dividend. $$ \left(y^{3}-64\right) \div(y+4) $$

Rewrite each equation in vertex form. $$ y=-2 x^{2}+x-10 $$

Exponents that are irrational numbers can be defined so that all the properties of rational exponents are also true for irrational exponents. Use those properties to simplify each expression. $$5^{2 \sqrt{3}} \cdot 25^{-\sqrt{3}}$$

Rationalize the denominator of each expression. Assume that all variables are positive. \(\frac{\sqrt[3]{x}}{\sqrt[3]{3 y}}\)

Find each indicated root if it is a real number. $$ \sqrt[5]{-243} $$

Solve each equation by factoring. \(x^{2}-7 x+12=0\)

Let \(f(x)=-4 x+1\) and \(g(x)=2 x-6 .\) Find \((g-f)(x)\) $$\begin{array}{llll}{\text { A. } 6 x-5} & {\text { B. } 6 x-7} & {\text { C. }-6 x+5} & {\text { D. }-6 x+7}\end{array}$$

Divide. Tell whether each divisor is a factor of the dividend. $$ \left(x^{3}+27\right) \div(x+3) $$

Rewrite each equation in vertex form. $$ y=\frac{x^{2}}{4}+2 x-1 $$

Exponents that are irrational numbers can be defined so that all the properties of rational exponents are also true for irrational exponents. Use those properties to simplify each expression. $$\frac{1}{9^{\frac{1}{\sqrt{2}}}}$$

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

Find each indicated root if it is a real number. $$ \sqrt[3]{0.064} $$

Solve each equation by factoring. \(x^{2}-8 x+15=0\)

If \(f(x)=2 x^{2}\) and \(g(x)=3 x,\) what is \((g \circ f)(x) ?\) $$\begin{array}{llll}{\text { F. } 6 x^{2}} & {\text { G. } 9 x^{2}} & {\text { H. } 18 x^{2}} & {\text { J. } 8 x^{4}}\end{array}$$

Divide. Tell whether each divisor is a factor of the dividend. $$ \left(6 a^{3}+a^{2}-a+4\right) \div(2 a+1) $$

Exponents that are irrational numbers can be defined so that all the properties of rational exponents are also true for irrational exponents. Use those properties to simplify each expression. $$\left(3^{2+\sqrt{2}}\right)^{2-\sqrt{2}}$$

Solve using the Quadratic Formula. \(5 x^{2}+x=3\)

Find each indicated root if it is a real number. $$ \sqrt[4]{810,000} $$

Let \(f(x)=2 x-3\) and \(g(x)=-x^{2}-1 .\) Find \((g \circ f)(x)\) $$\begin{array}{ll}{\text { A. }-2 x^{3}+3 x^{2}-2 x+3} & {\text { B. }-4 x^{2}+12 x-10} \\ {\text { C. }-x^{2}+2 x-4} & {\text { D. }-x^{2}-2 x+2}\end{array}$$

Solve each equation by factoring. \(x^{2}+9 x+20=0\)

Divide. Tell whether each divisor is a factor of the dividend. $$ \left(6 a^{3}+a^{2}-a+4\right) \div(2 a+1) $$

Weather Using data for the effect of temperature and wind on an exposed face, the National Weather Service uses the following formula. Wind Chill Index \(=35.74+0.6215 T-35.75 V^{0.16}+0.4275 T V^{0.16}\) \(T\) is the temperature in degrees Fahrenheit and \(V\) is the velocity of the wind in miles per hour. Frostbite occurs in about 15 minutes when the wind chill index is about \(-20 .\) Find the wind speed that produces a wind chill index of \(-20\) when the temperature is \(5^{\circ} \mathrm{F}\) .

Solve using the Quadratic Formula. \(3 x^{2}+9 x=27\)

List all possible rational roots for each equation. Then use the Rational Root Theorem to find each root. $$ 2 x^{3}+3 x^{2}-8 x-12=0 $$