Chapter 6

Perform the operations and simplify the result when possible. Be careful to apply the correct method, because these problems involve addition, subtraction, multiplication, and division of rational expressions. $$\frac{z^{2}-9}{z^{2}+4 z+3} \div \frac{z^{2}-3 z}{(z+1)^{2}}$$

Simplify each expression. \(-2\left(3 y^{3}-2 y+7\right)-\left(y^{2}+2 y-4\right)+4\left(y^{3}+2 y-1\right)\)

Explain how to multiply two rational expressions.

An electric company charges \(\$ 7.50\) per month plus \(9 \notin\) for each kilowatt hour ( \(\mathrm{kwh}\) ) of electricity used. a. Find a linear function that gives the total cost of \(n\) kwh of electricity. (Hint: See Example 1.) b. Find a rational function that gives the average cost per \(\mathrm{kwh}\) when using \(n\) kwh. c. Find the average cost per kwh when 775 kwh are used.

Suppose that \(P(x)=x^{100}-x^{99}+x^{98}-x^{97}+\cdots+x^{2}-x+1\). Find the remainder when \(P(x)\) is divided by \(x+1\)

Explain what it means to clear a rational equation of fractions. Give an example.

Perform the operations and simplify the result when possible. Be careful to apply the correct method, because these problems involve addition, subtraction, multiplication, and division of rational expressions. $$\frac{1}{m+1}+\frac{1}{m-1}+\frac{2}{m^{2}-1}$$

Simplify each expression. \(3\left(4 y^{3}+3 y-2\right)+2\left(3 y^{2}-y+3\right)-5\left(2 y^{3}-y^{2}-2\right)\)

Write some comments to the student who wrote the following solution, explaining the error. $$ \begin{aligned} \frac{x^{2}+x-2}{x^{2}-4} \cdot \frac{x-2}{x-1} &=\frac{(x+2)(x-1)(x-2)}{(x+2)(x-2)(x-1)} \\ &=0 \end{aligned} $$

The rational function $$ f(t)=\frac{t^{2}+3 t}{2 t+3} $$ gives the number of hours it would take two pipes, working together, to fill a pool that the larger pipe (working alone) could fill in \(t\) hours and the smaller pipe (working alone) could fill in \(t+3\) hours. a. If the smaller pipe could fill a pool in 7 hours, how long would it take both pipes to fill the pool? b. If the larger pipe could fill a pool in 8 hours, how long would it take both pipes to fill the pool?

Would you use the same approach to answer the following problems? Explain why or why not. Simplify: \(\frac{x^{2}-10}{x^{2}-1}-\frac{3 x}{x-1}-\frac{2 x}{x+1}\) Solve: \(\frac{x^{2}-10}{x^{2}-1}-\frac{3 x}{x-1}=-\frac{2 x}{x+1}\)

Solve each problem by writing a variation model. Electronics. The resistance of a wire is directly proportional to the length of the wire and inversely proportional to the square of the diameter of the wire. If the resistance is 11.2 ohms in a 80 -foot-long wire with diameter 0.01 inch, what is the resistance in a 160 -foot-long wire with diameter 0.04 inch?

Perform the operations and simplify the result when possible. Be careful to apply the correct method, because these problems involve addition, subtraction, multiplication, and division of rational expressions. $$\frac{27 p^{4}}{35 q} \div \frac{9 p}{21 q}$$

Perform each division. \(\left(3 c^{2}-\frac{7}{4} c-3\right) \div(4 c+3)\)

The rational function $$ f(t)=\frac{t^{2}+2 t}{2 t+2} $$ gives the number of days it would take two webpage designers, working together, to design a standard website for a business that designer 1 (working alone) could complete in \(t\) days and designer 2 (working alone) could complete in \(t+2\) days. a. If designer 1 could complete the website in 15 days, how long would it take both designers working together? b. If designer 2 could complete a website in 20 days, how long would it take both designers working together?

Solve each problem by writing a variation model. Business Models. A businessman who sells widgets has found that the revenue from their sale varies directly as the advertising budget and inversely as the price. When \(\$ 105,000\) was spent on advertising and the widgets were priced at \(\$ 19.95,\) the revenue from their sale was \(\$ 200,000\) How many widgets would he expect to sell at \(\$ 17.50\) each if \(\$ 700,000\) was spent on advertising?

Perform the operations and simplify the result when possible. Be careful to apply the correct method, because these problems involve addition, subtraction, multiplication, and division of rational expressions. $$\frac{12 t}{25 s^{5}} \div \frac{10 t}{15 s^{2}}$$

Perform each division. Divide \(x^{5}-1\) by \(x-1\)

Write italicized number in scientific notation. Oil. The total cost of the Alaskan pipeline, running 800 miles from Prudhoe Bay to Valdez, was \(\$ 9,000,000,000\)

Solve each problem by writing a variation model. Tension in a String. When playing with a Skip It toy, a child swings a weighted ball on the end of a string in a circular motion around one leg while jumping over the revolving string with the other leg. See the illustration. The tension \(T\) in the string is directly proportional to the square of the speed \(s\) of the ball and inversely proportional to the radius \(r\) of the circle. If the tension in the string is 6 pounds when the speed of the ball is 6 feet per second and the radius is 3 feet, find the tension when the speed is 8 feet per second and the radius is 2.5 feet.

Perform the operations and simplify the result when possible. Be careful to apply the correct method, because these problems involve addition, subtraction, multiplication, and division of rational expressions. $$\frac{6}{5 d^{2}-5 d}-\frac{3}{5 d-5}$$

Perform each division. \(\frac{c^{4}-c^{2} d^{2}+10 c^{2}-6 d^{2}+23}{c^{2}+6}\)

Complete the rules for exponents. Assume that there are no divisions by 0. a. \(x^{m} x^{n}=\) b. \(\left(x^{m}\right)^{m}=\) c. \((x y)^{n}=\) d. \(\left(\frac{x}{y}\right)^{n}=\) e. \(x^{0}=\) f. \(x^{-n}=\)

Solve each problem by writing a variation model. Gas Pressure. The pressure of a certain amount of gas is directly proportional to the temperature (measured on the Kelvin scale) and inversely proportional to the volume. A sample of gas at a pressure of 1 atmosphere occupies a volume of 1 cubic meter at a temperature of 273 Kelvin. When heated, the gas expands to twice its volume, but the pressure remains constant. To what temperature is it heated?

Perform the operations and simplify the result when possible. Be careful to apply the correct method, because these problems involve addition, subtraction, multiplication, and division of rational expressions. $$\frac{9}{2 r^{2}-2 r}-\frac{5}{2 r-2}$$

Perform each division. \(\frac{0.03 a^{2}+0.17 a+0.1}{0.03 a+0.02}\) (Hint: Think of a way to simplify the division.)

Complete the rules for exponents. Assume that there are no divisions by 0. a. \(\frac{x^{m}}{x^{n}}=\) b. \(\left(\frac{x}{y}\right)^{-n}=(\quad)\) c. \(\frac{x^{-m}}{y^{-n}}=\) d. \(x^{1}=\)

What does it mean when we say that \(\frac{3 x-12}{3 x+15}\) and \(\frac{x-4}{x+5}\) are equivalent expressions?

Write italicized number in scientific notation. Radioactivity. The least stable radioactive isotope is lithium \(5,\) which decays in 0.00000000000000000000044 second.

Explain the difference between a ratio and a proportion.

Perform the operations and simplify the result when possible. Be careful to apply the correct method, because these problems involve addition, subtraction, multiplication, and division of rational expressions. $$\frac{s^{3} t}{4 s^{2}-9 t^{2}} \cdot \frac{4 s^{2}-12 s t+9 t^{2}}{s^{3} t^{2}}$$

Insert either a multiplication symbol \(\cdot\) or a division symbol \(\div\) in each blank to make a true statement. $$ \frac{x^{2}}{y} \quad \frac{x}{y^{2}} \quad \frac{x^{2}}{y^{2}}=\frac{x^{3}}{y} $$

Give examples of two quantities from everyday life that vary directly and two quantities that vary inversely.

Perform the operations and simplify the result when possible. Be careful to apply the correct method, because these problems involve addition, subtraction, multiplication, and division of rational expressions. $$\frac{25 x^{2}-40 x y+16 y^{2}}{x^{2} y^{4}} \cdot \frac{x y^{4}}{25 x^{2}-16 y^{2}}$$

Perform each division. \(2 . 5 x - 3 . 7 \sqrt { - 2 2 . 2 5 x ^ { 2 } - 3 8 . 9 x - 1 6 . 6 5 }\)

Solve: \(\left(\frac{1}{2}\right)^{-1}=\frac{5 b^{-1}}{2}+2 b(b+1)^{-1}\)

Perform the indicated operations. $$ \left(\frac{5}{2} w^{3}+\frac{1}{4} w^{2}+\frac{3}{5}\right)-\left(\frac{1}{3} w^{3}+\frac{1}{2} w^{2}-\frac{1}{5}\right) $$

Perform the operations and simplify the result when possible. Be careful to apply the correct method, because these problems involve addition, subtraction, multiplication, and division of rational expressions. $$\frac{4}{x^{2}-2 x-3}-\frac{x}{3 x^{2}-7 x-6}$$

Perform each operation. $$ \left(a^{2}-4 a-3\right)(a-2) $$

Write a rational equation that has an extraneous solution of \(3 .\)

Perform the indicated operations. $$ \left(6 a^{2} x^{3}-2 a x^{2}+3 a^{3}\right)+\left(-4 a^{2} x^{3}-2 a^{3}\right) $$

Perform the operations and simplify the result when possible. Be careful to apply the correct method, because these problems involve addition, subtraction, multiplication, and division of rational expressions. $$\frac{x+3}{2 x^{2}-5 x+2}-\frac{3 x-1}{x^{2}-x-2}$$

Perform each operation. $$ \left(3 c^{2}+5 c\right)+\left(7-c^{2}-5 c\right) $$

Let \(f(x)=\frac{x^{3}-3 x^{2}+12}{x} .\) For what values of \(x\) is \(f(x)=4 ?\)

Perform the indicated operations. $$ (3 y+1)\left(2 y^{2}+3 y+2\right) $$

Perform the operations and simplify the result when possible. Be careful to apply the correct method, because these problems involve addition, subtraction, multiplication, and division of rational expressions. $$\begin{aligned}&\text { Let } f(x)=\frac{3 x}{x^{2}-25} \text { and } g(x)=\frac{4}{x+5}\\\ &\text { Find } f(x)+g(x)\end{aligned}$$

Perform each operation. $$ -3 m n^{2}\left(m^{3}-7 m n-2 m^{2}\right) $$

Let \(f(x)=\frac{x^{3}+2 x^{2}-32}{x} .\) For what values of \(x\) is \(f(x)=16 ?\)

Perform the indicated operations. $$ \left(5 k-6 m^{2}\right)^{2} $$

Perform the operations and simplify the result when possible. Be careful to apply the correct method, because these problems involve addition, subtraction, multiplication, and division of rational expressions. $$\begin{aligned}&\text { Let } f(x)=\frac{5}{2 x-4} \text { and } g(x)=\frac{3}{2-x}\\\ &\text { Find } f(x)-g(x)\end{aligned}$$