Exam 3 Review
(7.1-7.5, 8.1-8.5)
1. Simplify the rational expression. Write your answer with only positive exponents.
(a) 64−2/3
9
𝑥 2/3 ∙𝑥 −5/6
(b)
𝑥 1/9
2. Simplify the radical expression. Assume all variables are positive. If the expression is
not real, write “Not a real number.”
3
(a) −8𝑎21 𝑏 6
(b) −16𝑥 4 𝑦 2
3. Simplify. Assume all variables are positive. Do not leave your answer with any radicals
in the denominator.
(a) 7𝑏 𝑎5 𝑏 3 − 𝑎𝑏 4𝑎3 𝑏 3
3
(b) 54𝑎8 𝑏11 𝑐15
3
(c) 3
4𝑥
2+ 3
(d)
3− 2
4. Multiply.
(a) 3𝑥 27𝑥 − 12
(b)
2𝑥 − 4
2
5. State the domain of the function in set-builder notation. Then plot points and sketch
the graph. You must label 3 points on the graph.
(a) 𝑓 𝑥 = − 𝑥 + 2 1/2
3
(b) 𝑔 𝑥 = 𝑥 − 1
6. Solve the radical equation. It may be necessary to check your answer(s).
3
(a) 2𝑥 − 3 + 5 = 2
(b) 𝑥 + 5 = 5 − 𝑥
(c) 8𝑥 = −8
7. An object is dropped from a bridge. Find the distance the object has fallen when the
speed reaches 80 ft/s. Use the equation = 8 𝑑 , where v is the speed of the object and
d is the distance.
8. Multiply. Write your answers in complex number standard form 𝑎 + 𝑏𝑖.
(a) −2 18 − −8
(b) 4 − 3𝑖 2
9. Simplify. Write your answers in complex number standard form 𝑎 + 𝑏𝑖.
4−6𝑖
(a)
−2𝑖
(b)
2−3𝑖
3+𝑖
2
10. Write a quadratic equation that has integer coefficients with solutions 2 and − 3 .
11. Solve by taking square roots. Answers may be complex. Simplify your answers.
(a) 𝑢2 − 54 = 0
(b) 2 𝑥 − 2 2 = −8
12. Solve by completing the square. Answers may be complex. Simplify your answers.
4𝑥 2 − 4𝑥 + 17 = 0
13. Solve by using the quadratic formula. Answers may be complex. Simplify your answers.
2𝑥 2 − 8𝑥 + 5 = 0
14. Solve. Answers may be complex. Simplify your answers. It may be necessary to check
your answers.
(a) 𝑥 4 + 5𝑥 2 − 36 = 0
(b) 𝑥 − 2𝑥1/2 − 8 = 0
15. The height of a triangle is 3 inches less than the base of the triangle. The area of the
triangle is 80 inches 2 . Find the height of the triangle. Round to the nearest hundredth.
16. A diver jumps from a platform that is 10 m high. The height of the diver 𝑡 seconds after
the beginning of the dive is given by 𝑕 = −4.9𝑡 2 + 3.2𝑡 + 10.5 . To the nearest
hundredth of a second, how long after the beginning of the dive will the diver enter the
water?
(Hint: The diver’s height is 0 m when entering the water)
17. Solve and graph the solution set. Write the solution in set-builder notation.
𝑥+3 (𝑥−1)
(a)
≥0
𝑥−2
(b) 𝑥 3 + 3𝑥 2 − 𝑥 − 3 < 0