124b End of Semester Practice Problems. Simplify the radical. 1) ) ) ) 4) ) 5) 5 (-3)5 5)

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1 124b End of Semester Practice Problems Name Simplify the radical. 1) ) 2) ) 3) ) 4) ) 5) 5 (-3)5 5) 1

2 6) 3 (x + 2)3 6) Evaluate the expression, if possible. 7) 641/2 7) 8) /3 8) Rewrite the expression with a positive rational exponent. Simplify, if possible. 9) y 3/4 y1/4 9) 10) 2-2/7 23/7 10) 2

3 Simplify by factoring out the given factor. 11) 8k-3 + k-7; k-7 11) Use the product rule to multiply. Assume all variables represent positive real numbers. 12) ) 13) ) Use the product rule to simplify the expression. Assume that the variables can be any real number. 14) ) 15) x24 15) 16) 112 x 2y 16) 3

4 Use the quotient rule to divide and simplify. 8 17) 25 17) 18) ) 19) 63x11 7x 19) Add or subtract. Assume all variables represent positive real numbers. 20) ) 21) ) 4

5 22) ) Rationalize the denominator. Assume that all variables represent positive real numbers. 4 23) 13 23) 24) ) 25) ) Rationalize the denominator and simplify. Assume that all variables represent positive real numbers. 6 26) 26) 2-9 5

6 27) ) 28) ) 29) z + z m 29) Evaluate the radical function at the indicated value. 30) f(x) = - 2x - 6 f(21) 30) Find the domain of the given function. 31) f(x) = 8-7x 31) 6

7 32) f(x) = 3 5-4x 32) 33) f(x) = 4 10 x ) Determine the domain and range of the function. Then graph it. 34) f(x) = x ) Use the square root property to solve the equation. 35) x2 = ) 7

8 36) x2 = 15 36) 37) 5x2 = ) 38) (x - 7)2 = 36 38) 39) (x + 8)2 = 20 39) Complete the square for the binomial. Then factor the resulting perfect square trinomial. 40) x2-16x 40) 41) x2-7x 41) 8

9 Solve the equation by completing the square. 42) x2-2x - 8 = 0 42) 43) x2 + 3x - 9 = 0 43) 44) x2 + x + 9 = 0 44) 45) 4x2 + 1 = 3x 45) Use the quadratic formula to solve the equation. 46) x2 + 6x + 9 = 0 46) 47) 2x2-5x - 7 = 0 47) 9

10 48) 2x2 = -8x ) 49) 16x2 + 1 = 7x 49) Determine the discriminant of the quadratic equation. Use the value of the discriminant to determine whether the quadratic equation has two rational solutions, two irrational solutions, one repeated real solution, or two complex solutions that are not real. 50) x2 + 12x + 36 = 0 50) 51) x2-3x + 3 = 0 51) 52) 36x2-12x = 0 52) 53) 4 + 5x2 = 5x 53) 10

11 Solve the problem. 54) The hypotenuse of an isosceles right triangle is 2 feet longer than either of its legs. Find the exact length of each side. 54) 55) The area of a rectangular wall in a classroom is 260 square feet. Its length is 4 feet shorter than three times its width. Find the length and width of the wall of the classroom. 55) Solve the equation. 56) x4-40x = 0 56) 57) x - 12 x + 35 = 0 57) 58) (2x - 4)2 + 2(2x - 4) - 3 = 0 58) 11

12 For the given functions f and g, find the indicated value. 59) f(x) = x2 + 6x, g(x) = x + 2 (f g)(4) 59) 60) f(x) = 4x + 6, g(x) = 4x2 + 3 (f g)(0) 60) 61) f(x) = 3x3, g(x) = -x2 + 4 (g g)(3) 61) Indicate whether the function is one-to-one. 62) Town Number of Commissioners Winthrop > 11 Lincolnshire > 16 Baker > 16 Middleville > 9 62) Find the inverse of the function. 63) {(7, 1), (5, 2), (3, 3), (1, 4)} 63) 12

13 Indicate whether the graph represents a one-to-one function. 64) 64) Use the graph of the given one-to-one function to sketch the graph of the inverse function. For convenience, the graph of y = x is also given. 65) 65) Find the inverse of the one-to-one function. 66) f(x) = (x + 6)3 66) 13

14 67) f(x) = 2x 67) 68) f(x) = 5 x 68) 69) f(x) = 3 x ) Determine whether the functions f and g are inverses of each other. 70) f(x) = 8 x - 6 ; g(x) = 8 x ) Approximate the value using a calculator. Express answer rounded to three decimal places. 71) 2e 71) 14

15 Graph the exponential function. 72) f(x) = 3 x ) 73) f(x) = ( 1 2 )x 73) 15

16 74) y = -3x 74) Solve the equation. 75) 3x = ) 76) 5-x = ) 77) 2x = ) 16

17 78) 1 3 x - 9 = 0 78) 79) 3(x2-12) = 81x 79) Solve the problem. 80) The expected future population of a small town, which currently has 7900 residents, can be approximated by the formula y = 7900(1.6)-0.3t where t is the number of years in the future. Find the expected population of the town 40 years in the future. 80) Change the exponential expression to an equivalent expression involving a logarithm. 81) 4-3 = ) 82) 8 x = 64 82) 17

18 83) ex = 6 83) Change the logarithmic expression to an equivalent expression involving an exponent. 1 84) log = -3 84) ) log 7 49 = x 85) 86) logb 49 = ) Evaluate the expression without using a calculator. 87) log ) 88) log ) 18

19 89) log ) 90) log ) Find the domain of the logarithmic function. 91) f(x) = log(x - 4) 91) Use a calculator to approximate the logarithm to four decimal places. 92) log ) Use a calculator to approximate the natural logarithm to four decimal places. 93) ln 5 93) Solve. 94) log = x 94) 19

20 95) log 2 1 = x 95) 96) log = y 96) 97) log 6 (x - 4) = 2 97) 98) ln(x) = 4 98) Solve the problem. 99) The Richter Scale measures the magnitude M of an earthquake. An earthquake whose seismographic reading measures x millimeters 100 kilometers from the epicenter has x magnitude M given by M(x) = log. Give the magnitude of an earthquake that 10-3 resulted in a seismographic reading of 69,003 millimeters 100 kilometers from its epicenter. 99) 20

21 Use the properties of logarithms to find the exact value of the expression. Do not use a calculator. 100) ln e ) 101) log log ) 102) eln 6 102) 103) 10log 18 - log 3 103) Write as the sum and/or difference of logarithms. Express powers as factors. 104) log b xy3 z5 104) 105) log r s 105) 21

22 106) log 5 x ) Write the expression as a logarithm of a single expression. Assume that variables represent positive numbers. 107) log log ) 108) log log log ) 109) 4 log b m - log b n 109) 110) 1 2 (log 7 (r - 1) - log7 r) 110) Use the Change-of-Base Formula and a calculator to evaluate the logarithm. Round to four decimal places. 111) log ) 22

23 112) log 1/ ) Solve the equation. 113) log 3 (x + 5) + log 3 (x - 1) = 3 113) 114) 1 2 log 7 x = log ) 115) log 3 (x - 5) + log 3 (x - 11) = 3 115) 116) log x + log (x - 1) = log ) Solve the equation. Give an exact solution. 117) 3x + 8 = 2 117) 23

24 118) e(x + 8) = 5 118) 119) 47x = ) Solve the equation. Give an approximate solution to four decimal places. 120) 2x + 6 = 3 120) 121) e(x + 6) = 4 121) Solve the problem. 122) The value V of a car that is t years old can be modeled by V(t) = 19,655(0.83)t. According to the model, when will the car be worth $6000? 122) 123) The formula P = 112e0.028t models the population of a particular city, in thousands, t years after When will the population of the city reach 125 thousand? 123) 24

25 Find the distance d(p1, P2) between the points P1and P2. 124) P1 = (-4 7, 3); P2 = (-3 7, 6) 124) Find the midpoint of the line segment formed by joining the points P1 and P2. 125) P1 = (3, 1); P2 = (-5, 6) 125) Find the center and radius of the circle whose graph is shown. Write the standard form of the equation of the circle. 126) 126) Write the standard form of the equation of the circle whose radius is r and whose center is (h, k). 127) r =8; (h, k) = (0, 7) 127) 25

26 Find the standard form of the equation of the circle. 128) Center at (5, -6) and containing the point (3, 2). 128) Find the center (h, k) and the radius r of the circle. 129) x2 + y2-4x - 12y + 25 = 0 129) Write the first four terms of the sequence. 130) {4n - 2} 130) 131) 3 5 n 131) The given pattern continues. Write down the nth term of the sequence suggested by the pattern. 132) 0, 2, 6, 12, 20, ) 26

27 133) 1 1, 1 4, 1 9, 1 16, 1, ) 25 Write out the sum. Do not evaluate. 134) 6 (5k - 3) k = 1 134) Find the indicated sum. 135) 5 (4i - 4) 135) i = 2 136) 4 i = 1 1 3i 136) 137) 4 i = 1 2 i 137) 27

28 138) i 138) i = 1 Express the sum using summation notation. Use 1 as the lower limit of summation and i for the index of summation. 139) ) 140) ) Express the sum using summation notation. Use a lower limit of summation not necessarily 1 and k for the index of summation. 141) ) 142) ) 28

29 Determine whether the sequence is arithmetic. If the sequence is arithmetic, determine the first term a and common difference d. 143) 4, 12, 36, 108, 972, ) 144) 2, -1, -4, -7, -10, ) An arithmetic sequence is given. Find the common difference and write out the first four terms. 145) {7n + 6} 145) Find a formula for the nth term of the arithmetic sequence whose first term a and common difference d are given. 146) a = 20, d = ) Write a formula for the nth term of the arithmetic sequence. Use the formula to find the 20th term of the sequence. 147) 10, 1, -8, -17, ) 29

30 148) a = 3 5, d = ) Evaluate the given binomial coefficient ) 5 149) Expand the expression using the Binomial Theorem. 150) (x - 2)5 150) 151) (x2 + 5y)4 151) MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the term indicated in the expansion. 152) (x2 + y4) 9 ; 3rd term 152) A) 25920x14y8 B) 36x9y6 C) 36x14y8 D) 25920x9y6 30

Mock Final Exam Name. Solve and check the linear equation. 1) (-8x + 8) + 1 = -7(x + 3) A) {- 30} B) {- 6} C) {30} D) {- 28}

Mock Final Exam Name. Solve and check the linear equation. 1) (-8x + 8) + 1 = -7(x + 3) A) {- 30} B) {- 6} C) {30} D) {- 28} Mock Final Exam Name Solve and check the linear equation. 1) (-8x + 8) + 1 = -7(x + 3) 1) A) {- 30} B) {- 6} C) {30} D) {- 28} First, write the value(s) that make the denominator(s) zero. Then solve the