Problems

1. (a). The function $h$ is defined as follows.

   $h(x)=\dfrac{x-2}{x^2+2x-35}$

   Find $h(-4)$.

   (b). The function $f$ is defined as follows.

   $f(x)=\dfrac{x-2}{x^2+2x-35}$

   Find $f(5)$.

2. (a). The function $f$ is defined below.Find all values of $x$ that are NOT in the domain of $f$.

   $f(x)=\dfrac{x^2-3x-10}{x^2-9}$

   (b). The function $g$ is defined below. Find all values of $x$ that are NOT in the domain of $g$.

   $\dfrac{x+1}{x^2-7x-8}$

3. Simplify.

   (a). $\dfrac{49x^2-4}{7x^2+5x-2}$

   (b). $\dfrac{3x^2+10x+3}{3x^2-8x-3}$

4. Simplify.

   (a). $\dfrac{9x}{x^2-10x+9}\div \dfrac{x^2-8x}{x^2-81}\cdot \dfrac{x-8}{x+9}$

   (b). $\dfrac{x^2+x-12}{7x}\div \dfrac{x^2-9}{x^2+x}\div \dfrac{x+4}{x+3}$

5. (a). Add. Simplify your answer as much as possible.

   $\dfrac{x+3}{x-7}+\dfrac{x+6}{x-1}$

   (b). Subtract. Simplify your answer as much as possible.

   $\dfrac{x+3}{x+2}-\dfrac{x-4}{x}$

6. (a). Simplify.

   $\dfrac{\dfrac{8}{u+2}+\dfrac{1}{u-7}}{\dfrac{9}{u+2}}$

   (b). Simplify.

   $\dfrac{\dfrac{5}{x-2}-\dfrac{1}{x+6}}{\dfrac{6}{x+6}-\dfrac{2}{x-2}}$

7. (a). Solve for $n$

   $\dfrac{9}{n}+\dfrac{5}{m}=\dfrac{1}{p}$

   (b). Solve for $x$

   $\dfrac{7}{v}+\dfrac{2}{x}=y$

8. (a). Solve for $x$

   $\dfrac{2}{x}=\dfrac{x}{3x+8}$

   (b). Solve for $x$

   $\dfrac{x}{x-6}=\dfrac{2}{x-5}$

9. (a). The ratio of men to women working for a company is 4 to 3. If there are 91 employees total, how many men work for the company?

   (b). Each marble bag sold by Pablo's Marble Company contains 7 blue marbles for every 4 orange marbles. If a bag has 63 blue marbles, how many orange marbles does it contain?

10. (a). One hose can fill a small swimming pool in 95 minutes. A larger hose can fill the pool in 30 minutes. How long will it take the two hoses to fill the pool working together?

    (b). Working together, it takes two computers 10 minutes to send out a company's email. If it takes the slower computer 35 minutes to do the job on its own, how long will it take the faster computer to do the job on its own?

11. (a). Manuel swam 2 kilometers against the current in the same amount of time it took him to swim 8 kilometers with the current. The rate of the current was 3 kilometers per hour. How fast would Manuel swim if there were no current (in still water)?

    (b). A passenger train traveled 110 miles in the same amount of time it took a freight train to travel 90 miles. The rate of the freight train was 10 miles per hour slower than the rate of the passenger train. Find the rate of the passenger train.

12. Simplify each expression. Assume that the variables represent any real numbers.

    (a). $\sqrt{x^{16}}$

    (b). $\sqrt{u^{22}}$

13. (a). Find the domain of the function. Write your answer using interval notation.

    $g(x)=\sqrt{x}-8$

    (b). Find the domain of the function. Write your answer using interval notation.

    $h(x)=\sqrt{x+15}$

14. (a). Simplify the expression. Write your answer using only positive exponents. Assume that all variables are positive real numbers.

    $\dfrac{ b^{-\dfrac{1}{2}} \: b^{\dfrac{1}{3}} }{b^\dfrac{1}{4}}$

    (b). Simplify the expression. Write your answer using only positive exponents. Assume that all variables are positive real numbers.

    $\dfrac{ z^{\dfrac{1}{3}} } {z^{-\dfrac{3}{2}} \: z^{\dfrac{1}{2}}} $

15. (a). Simplify. Assume that all variables represent positive real numbers.

    $\sqrt{50x^9y^5}$

    (b). Simplify. Assume that all variables represent positive real numbers.

    $\sqrt{24y^2z^7}$

16. Write the following expressions in simplified radical form. Assume that all of the variables in the expressions represent positive real numbers.

    (a). $\sqrt[3]{72x^3z^{14}}$

    (b). $\sqrt[5]{160x^8w^{15}}$

    (c). $\sqrt[4]{96x^{16}w^{6}}$

17. Simplify. Assume that the variables represent positive real numbers.

    (a). $4y\sqrt{80y^3}+\sqrt{45y^5}$

    (b). $15\sqrt{w^7}-13w^3\sqrt{w}$

18. (a). Multiply. Simplify your answer as much as possible.

    $(5\sqrt{6}-1)(2+\sqrt{10})$

    (b). Multiply. Simplify your answer as much as possible.

    $(4\sqrt{5}+5\sqrt{6})(7\sqrt{5}-2\sqrt{6})$

19. Write in simplified radical form with at most one radical. Assume that the variable represents a positive real number.

    (a). $\sqrt[5]{y^3}\cdot \sqrt[3]{y}$

    (b). $\dfrac{\sqrt[6]{x^5}}{\sqrt{x}}$

20. Rationalize the denominator and simplify. Assume that the variable represents a positive real number.

    (a). $\dfrac{-9}{4\sqrt{u}-2}$

    (b). $\dfrac{7}{5+2\sqrt{w}}$

21. Rationalize the denominator and simplify.

    (a). $\sqrt[4]{\dfrac{5}{2}}$

    (b). $\dfrac{\sqrt[3]{3}}{\sqrt[3]{16}}$

22. (a). Solve for $u$, where $u$ is a real number.

    $\sqrt{-2u+31}=u+2$

    (b). Solve for $w$, where $w$ is a real number.

    $w-2=\sqrt{5w-16}$

23. (a). Add. Write your answer as a complex number in standard form.

    $(-3+2i)+(-4+6i)$

    (b). Subtract. Write your answer as a complex number in standard form.

    $(5-i)-(6-5i)$

24. Multiply. Write your answer as a complex number in standard form.

    (a). $(-6+3i)(-1+i)$

    (b). $(5-6i)(4-2i)$

25. Divide. Write your answer as a complex number in standard form.

    (a). $\dfrac{-2+i}{-6-4i}$

    (b). $\dfrac{-3+2i}{2+3i}$

26. Simplify the complex numbers as much as possible.

    (a). $i^{91}$

    (b). $i^{78}$