, Solution manual for Finite Mathematics, 8th
Edition Stefan Waner
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, Solutions Section 0.1
Section 0.1
1. 2(4 + (−1))(2 ⋅ −4) 2. 3 + ([4 − 2] ⋅ 9)
= 2(3)(−8) = (6)(−8) = −48 = 3 + (2 × 9) = 3 + 18 = 21
3. 20∕(3 * 4) − 1 4. 2 − (3 * 4)∕10
= 20
12
− 1 = 53 − 1 = 2
3
= 2 − 1210
= 2 − 65 = 4
5
3 + ([3 + (−5)]) 12 − (1 − 4)
3−2×2 2(5 − 1) ⋅ 2 − 1
5. 6.
3 + (−2) 1 12 − (−3) 15
= = = −1 = = =1
3−4 −1 16 − 1 15
7. (2 − 5 * (−1))∕1 − 2 * (−1) 8. 2 − 5 * (−1)∕(1 − 2 * (−1))
2 − 5 ⋅ (−1) 5 ⋅ (−1)
= − 2 ⋅ (−1) =2−
1 1 − 2 ⋅ (−1)
2+5 −5 5 11
= +2=7+2=9 =2− +2=2+ =
1 1+2 3 5
2 × (−1) 2 10. 2 + 4 ⋅ 3 2 = 2 + 4 × 9 = 2 + 36 = 38
9. 2 ⋅ (−1) 2∕2 =
2
2×1 2
= = =1
2 2
11. 2 ⋅ 4 2 + 1 12. 1 − 3 ⋅ (−2) 2 × 2
= 2 × 16 + 1 = 32 + 1 = 33 = 1 − 3 × 4 × 2 = 1 − 24 = −23
= 3 2 + 2 2 + 1 = 9 + 4 + 1 = 14 = 2 (2
2−2)
= 2 4−2 = 2 2 = 4
13. 3^2+2^2+1 14. 2^(2^2-2)
3 − 2(−3) 2 1 − 2(1 − 4) 2
−6(4 − 1) 2 2(5 − 1) 2 ⋅ 2
15. 16.
3 − 2 × 9 3 − 18 1 − 2(−3) 2 1−2×9
= = = =
−6(3) 2 −6 × 9 2(4) 2 ⋅ 2 2 × 16 × 2
−15 5 1 − 18 17
= = = =−
−54 18 64 64
1 3 121 121
= =
17. 10*(1+1/10)^3 18. 121/(1+1/10)^2
= 10(1 + ) = 10(1.1) 3
10 !1 + "
1 2 1.1 2
= 10 × 1.331 = 13.31 10
121
= = 100
1.21
, Solutions Section 0.1
−2 ⋅ 3 2 8(1 − 4) 2
19. 3 20. −
[ −(4 − 1) 2 ] [ −9(5 − 1) 2 ]
−2 × 9 ⎤ −18 8×9 72
= 3⎡ = 3[ = −[ = −(
⎣⎢ −3 ⎦⎥
2 −9 ] −9 × 16 ] −144 )
=3×2=6 1 1
= −(− ) =
2 2
1 2 2 1 2 2 2
21. 3⎡1 − (− ) ⎤ + 1 22. 3⎡ − ( ) ⎤ + 1
⎣⎢ 2 ⎦⎥ ⎣⎢ 9 3 ⎦⎥
1 2 1 4 2
= 3[1 − ] + 1 = 3[ − ] + 1
4 9 9
3 2 9 −3 2 −1 2
= 3[ ] + 1 = 3[ ] + 1 = 3[ ] + 1 = 3[ ] + 1
4 16 9 3
27 43 1 3 4
= +1= =3 +1= +1=
16 16 9 9 3
1 2 2
= #2$ − 2 = − =0 = − # 21 $ = − 41 = −2
23. (1/2)^2-1/2^2 24. 2/(1^2)-(2/1)^2
1 1 1 2 2
2 4 4 12 1
25. 3 × (2 − 5) = 3*(2-5) 5
26. 4 +
9
= 4+5/9 or 4+(5/9)
3 4−1
2−5 3
27. = 3/(2-5) 28. = (4-1)/3
3−1 3
30. 3 +
8+6 2−9
29. = (3-1)/(8+6) = 3+3/(2-9)
3 − 18 − 6.
Note 3-1/8-6 is wrong, as it corresponds to
4+7 4×2
31. 3 −
8 2
= 3-(4+7)/8 32. = 4*2/(2/3) or (4*2)/(2/3)
!3"
2 3+𝑥
− 𝑥𝑦 2 = 2/(3+x)-x*y^2 34. 3 +
3+𝑥 𝑥𝑦
33. = 3+(3+x)/(x*y)
60 𝑥2 − 3
35. 3.1𝑥 3 − 4𝑥 −2 − 36. 2.1𝑥 −3 − 𝑥 −1 +
𝑥2 − 1 2
= 3.1x^3-4x^(-2)-60/(x^2-1) = 2.1x^(-3)-x^(-1)+(x^2-3)/2
, 2
Solutions Section 0.1
2
!3"
3
38.
5
37. !5"
= (2/3)/5 = 2/(3/5)
39. 3 4−5 × 6 2
3+5 7−9
40.
= 3^(4-5)*6
= 2/(3+5^(7-9))
Note that the entire exponent is in parentheses.
Note that the entire exponent is in parentheses.
4 1+4
−3 −3
41. 3 1 + 42. 3
( 100 ) ( 100 )
= 3*(1+4/100)^(-3) = 3((1+4)/100)^(-3)
43. 3 2𝑥−1 + 4 𝑥 − 1 2
44. 2 𝑥 − (2 2𝑥) 2
= 3^(2*x-1)+4^x-1 = 2^(x^2)-(2^(2*x))^2
2 2
45. 2 2𝑥 −𝑥+1 46. 2 2𝑥 −𝑥 + 1
= 2^(2x^2-x+1) = 2^(2x^2-x)+1
Note that the entire exponent is in parentheses. Note that the entire exponent is in parentheses.
4𝑒 −2𝑥 𝑒 2𝑥 + 𝑒 −2𝑥
2 − 3𝑒 −2𝑥 𝑒 2𝑥 − 𝑒 −2𝑥
47. 48.
= 4*e^(-2*x)/(2-3e^(-2*x)) = (e^(2*x)+e^(-2*x))/(e^(2*x)-
or 4(*e^(-2*x))/(2-3e^(-2*x)) e^(-2*x))
or (4*e^(-2*x))/(2-3e^(-2*x))
2 2 1 2 2 2
49. 3(1 − !− 12 " ) + 1 50. 3⎛ − ( ) ⎞ + 1
⎜⎝ 9 3 ⎟⎠
= 3(1-(-1/2)^2)^2+1 = 3(1/9-(2/3)^2)^2+1
, Solutions Section 0.2
Section 0.2
1. 3 3 = 27 2. (−2) 3 = −8
3. −(2 ⋅ 3) 2 = −(2 2 ⋅ 3 2) = −(4 ⋅ 9) = −36 4. (4 ⋅ 2) 2 = 4 2 ⋅ 2 2 = 16 ⋅ 4 = 64 or
(4 ⋅ 2) 2 = 8 2 = 64
−(2 ⋅ 3) 2 = −6 2 = −36
or
−2 2 (−2) 2 4 3 3 3 3 27
6. ( ) = 3 =
5. ( = =
3 ) 32 9 2 2 8
1 1 1 1 1
7. (−2) −3 = = =− 8. −2 −3 = − =−
(−2) 3 −8 8 23 8
1 −2 1 1 −2 −2 1 1 9
9. ( ) = = = 16 10. ( = = =
4 (1∕4) 2 1∕16 3 ) (−2∕3) 2 4∕9 4
11. 2 ⋅ 3 0 = 2 ⋅ 1 = 2 12. 3 ⋅ (−2) 0 = 3 ⋅ 1 = 3
13. 2 32 2 = 2 3+2 = 2 5 = 32 14. 3 23 = 3 23 1 = 3 2+1 = 3 3 = 27or
3 23 = 9 ⋅ 3 = 27
2 32 2 = 8 ⋅ 4 = 32
or
15. 2 22 −12 42 −4 = 2 2−1+4−4 = 2 1 = 2 1
16. 5 25 −35 25 −2 = 5 2−3+2−2 = 5 −1 =
5
17. 𝑥 3𝑥 2 = 𝑥 3+2 = 𝑥 5 18. 𝑥 4𝑥 −1 = 𝑥 4−1 = 𝑥 3
𝑦 20. −𝑥𝑦 −1𝑥 −1 = −𝑥 1−1𝑦 −1 = −𝑥 0𝑦 −1
19. −𝑥 2𝑥 −3𝑦 = −𝑥 2−3𝑦 = −𝑥 −1𝑦 = −
𝑥 1
=−
𝑦
𝑥3 1 𝑦5
= 𝑥 3−4 = 𝑥 −1 = = 𝑦 5−3 = 𝑦 2
𝑥4 𝑥 𝑦 3
21. 22.
𝑥 2𝑦 2 𝑥 −1𝑦 1
= 𝑥 2−(−1)𝑦 2−1 = 𝑥 3𝑦 = 𝑥 −1−2𝑦 1−2 = 𝑥 −3𝑦 −1 = 3
𝑥 −1𝑦 𝑥 2𝑦 2 𝑥 𝑦
23. 24.
, (𝑥𝑦 −1𝑧 3) 2 𝑥 2(𝑦 −1) 2(𝑧 3) 2 𝑥 2𝑦𝑧 2 𝑥 2𝑦𝑧 2
Solutions Section 0.2
= =
𝑥 2𝑦𝑧 2 𝑥 2𝑦𝑧 2 (𝑥𝑦𝑧 −1) −1 𝑥 −1𝑦 −1(𝑧 −1) −1
25. 26.
𝑥 𝑦 𝑧
2 −2 6
𝑥 2𝑦𝑧 2
= 2 2 = 𝑥 2−2𝑦 −2−1𝑧 6−2 = −1 −1 = 𝑥 2−(−1)𝑦 1−(−1)𝑧 2−1
𝑥 𝑦𝑧 𝑥 𝑦 𝑧
𝑧4 = 𝑥 3𝑦 2𝑧 4
= 𝑥 0𝑦 −3𝑧 4 = 3
𝑦
𝑥𝑦 −2𝑧 3 (𝑥𝑦 −2𝑧) 3 𝑥 2𝑦 −1𝑧 0 (𝑥 2𝑦 −1𝑧 0) 2
2
" = 28. ( =
𝑥 −1𝑧 (𝑥 −1𝑧) 3 𝑥𝑦𝑧 ) (𝑥𝑦𝑧) 2
27. !
𝑥 𝑦 𝑧
3 −6 3
𝑥 4𝑦 −2
= −3 3 = 𝑥 3−(−3)𝑦 −6𝑧 3−3 = 2 2 2 = 𝑥 4−2𝑦 −2−2𝑧 −2
𝑥 𝑧 𝑥 𝑦 𝑧
𝑥6 𝑥2
= 𝑥 6𝑦 −6𝑧 0 = = 𝑥 2𝑦 −4𝑧 −2 = 4 2
𝑦6 𝑦 𝑧
𝑥 −1𝑦 −2𝑧 2 𝑥𝑦 −2
−2 −3
29. ( ) = (𝑥 𝑦 𝑧 )
−1−1 −2−1 2 −2
= (𝑥 1−2𝑦 −2+1𝑧 −1) −3
𝑥𝑦 ( 𝑥 2𝑦 −1𝑧 )
30.
= (𝑥 −2𝑦 −3𝑧 2) −2 = 𝑥 4𝑦 6𝑧 −4 = (𝑥 −1𝑦 −1𝑧 −1) −3 = 𝑥 3𝑦 3𝑧 3
𝑥 4𝑦 6
= 4
𝑧
31. √4 = 2 32. √5 ≈ 2.236
1 √1 1 1 √1 1
= = = =
√4 √4 2 √9 √9 3
33. 34.
16 √16 4 9 √9 3
= = = =
√9 √9 3 √4 √4 2
35. 36.
√4 2 6 6
= =
5 5 √25 5
37. 38.
39. √9 + √16 = 3 + 4 = 7 40. √25 − √16 = 5 − 4 = 1
41. √9 + 16 = √25 = 5 42. √25 − 16 = √9 = 3
3 3 4 4
43. √8 − 27 = √−19 ≈ −2.668 44. √81 − 16 = √65 ≈ 2.839
3 3 3 3
3 √27 3 √8 × 64 = √8 ⋅ √64 =2⋅4=8
45. √27∕8 = =
2
46.
3
√8
, Solutions Section 0.2
47. √(−2) 2 = √4 = 2 48. √(−1) 2 = √1 = 1
1 16 √16 4 1 36 √36 6
(1 + 15) = = = =2 (3 + 33) = = = =2
√4 √4 √4 2 √9 √9 √9 3
49. 50.
51. √𝑎 2𝑏 2 = √𝑎 2√𝑏 2 = 𝑎𝑏 𝑎 2 √𝑎 2 𝑎
52. √ = =
𝑏 2 √𝑏 2 𝑏
53. √(𝑥 + 9) 2 = 𝑥 + 9
2
54. (√𝑥 + 9) = 𝑥 + 9
(𝑥 + 9 > 0 because 𝑥 is positive.)
3 3 3 4 𝑥4 4
55. √𝑥 3(𝑎 3 + 𝑏 3) = √𝑥 3 √(𝑎 3 + 𝑏 3) √𝑥 4 𝑥
56. √ = =
3
= 𝑥 √(𝑎 3 + 𝑏 3) (Not 𝑥(𝑎 + 𝑏)) 𝑎 4𝑏 4 4 4
√𝑎 4 √𝑏 4 𝑎𝑏
4𝑥𝑦 3 4𝑦 2 √4√𝑦 2 2𝑦 4(𝑥 2 + 𝑦 2) √ 4√ 𝑥 2 + 𝑦2
= √ 𝑥 = = =
√ 𝑥 2𝑦 √𝑥 √𝑥 √ 𝑐2 √𝑐 2
57. 58.
2√𝑥 2 + 𝑦 2
= (Not 2(𝑥 + 𝑦)∕𝑐)
𝑐
, Solutions Section 0.3
Section 0.3
3 1 −4 1
1. 3𝑥 −4 = 𝑥 = 4
𝑥4 2 2𝑥
2.
3 −2∕3 3 4 −3∕4 4
𝑥 = 𝑦 =
4 4𝑥 2∕3 5 5𝑦 3∕4
3. 4.
5. 1 − 0.3
− 65 𝑥 −1 = 1 − 0.3𝑥 2 − 5𝑥
6
1 0.1𝑥 −2 𝑥 4 0.1
𝑥 −2 + = + 2
3𝑥 −4 3 3 3𝑥
6.
3 5
7. 2 2∕3 = √2 2 8. 3 4∕5 = √3 4
3 3 4 4 4 4
10. 𝑦 7∕4 = √𝑦 7 = √𝑦 4 √𝑦 3 = |𝑦| √𝑦 3
3 3
9. 𝑥 4∕3 = √𝑥 4 = √𝑥 3 √𝑥 = 𝑥 √𝑥
11. (𝑥 1∕2𝑦 1∕3)
1∕5 5
= √√ 𝑥 √ 𝑦
3 𝑦 3∕2 √𝑦 3 |𝑦|√𝑦
12. 𝑥 −1∕3𝑦 3∕2 = = 3 = 3
𝑥 1∕3 √𝑥 √𝑥
3 3 3 4 3∕2 4√𝑥 3 4|𝑥|√𝑥
13. − 𝑥 −1∕4 = − =− 4 𝑥 = =
2 2𝑥 1∕4
2 √𝑥
14.
5 5 5
3 0.2 3𝑥 1∕2 3.1 11 −1∕7 11
15. 0.2𝑥 −2∕3 + = + − 𝑥 = 3.1𝑥 4∕3 −
7𝑥 −1∕2 𝑥 2∕3 7 𝑥 −4∕3 7 7𝑥 1∕7
16.
0.2 3√ 𝑥
3 11 3 11
= 3 + = 3.1 √𝑥 4 − 7 = 3.1𝑥 √𝑥 − 7
√𝑥 2 7 7 √𝑥 7 √𝑥
3 3 9 9(1 − 𝑥) 7∕3
= =
4(1 − 𝑥) 5∕2
4√(1 − 𝑥) 5 4(1 − 𝑥) −7∕3 4
17. 18.
3 3 3
9 √(1 − 𝑥) 7 9 √(1 − 𝑥) 6 √1 − 𝑥
3
=
4√(1 − 𝑥) 4√1 − 𝑥 = =
4 4
3
= 3
9(1 − 𝑥) 2 √1 − 𝑥
4(1 − 𝑥) 2√1 − 𝑥 =
4
19. √3 = 3 1∕2 20. √8 = 8 1∕2
21. √𝑥 3 = 𝑥 3∕2 3
22. √𝑥 2 = 𝑥 2∕3
, Solutions Section 0.3
3
23. √𝑥𝑦 2 = (𝑥𝑦 2) 1∕3 24. √𝑥 2𝑦 = (𝑥 2𝑦) 1∕2
𝑥2 𝑥2 𝑥 𝑥
= = 𝑥 2−1∕2 = 𝑥 3∕2 = = 𝑥 1−1∕2 = 𝑥 1∕2
√𝑥 𝑥 1∕2 √𝑥 𝑥 1∕2
25. 26.
3 3 2 2
= 𝑥 −2 = 𝑥3
5𝑥 2 5 5𝑥 −3 5
27. 28.
3𝑥 −1.2 1 3 1 2 𝑥 2.1 2 1.2 1 2.1
− 2.1 = 𝑥 −1.2 − 𝑥 −2.1 − = 𝑥 − 𝑥
2 3𝑥 2 3 3𝑥 −1.2 3 3 3
29. 30.
2𝑥 𝑥 0.1 4 4𝑥 2 𝑥 3∕2 2
− + 1.1 + − 2
3 2 3𝑥 3 6 3𝑥
31. 32.
2 1 4 4 1 2
= 𝑥 − 𝑥 0.1 + 𝑥 −1.1 = 𝑥 2 + 𝑥 3∕2 − 𝑥 −2
3 2 3 3 6 3
3√𝑥 5 4 3 5√𝑥 7
− + − + 3
4 3√𝑥 3𝑥√𝑥 5√𝑥 8 2 √𝑥
33. 34.
3𝑥 1∕2 5 4 3 5𝑥 1∕2 7
= − + = − +
4 3𝑥 1∕2 3𝑥 ⋅ 𝑥 1∕2 5𝑥 1∕2 8 2𝑥 1∕3
3 5 4 3 −1∕2 5 1∕2 7 −1∕3
= 𝑥 1∕2 − 𝑥 −1∕2 + 𝑥 −3∕2 = 𝑥 − 𝑥 + 𝑥
4 3 3 5 8 2
5 1 2 1 2
3 √𝑥 2 7 3𝑥 2∕5 7 − = −
− = − 8𝑥 𝑥 3 √𝑥 3 8𝑥
5 3∕2 3𝑥 3∕5
4 4 2𝑥 3∕2
√
2√𝑥 3
36.
35.
3 7 1 2
= 𝑥 2∕5 − 𝑥 −3∕2 = 𝑥 −3∕2 − 𝑥 −3∕5
4 2 8 3
1 3 3
3 √(𝑥 2 + 1) 7
− 2
(𝑥 2 + 1) 3 3 −
4 √𝑥 2 + 1 3(𝑥 2 + 1) −3 4
37.
1 3
38.
= − 2 2 3
= (𝑥 + 1) 3 − (𝑥 2 + 1) 7∕3
(𝑥 2 + 1) 3 (𝑥 2 + 1) 1∕3 3 4
3
= (𝑥 2 + 1) −3 − (𝑥 2 + 1) −1∕3
4
39. 4 −1∕24 7∕2 = 4 −1∕2+7∕2 = 4 6∕2 = 4 3 = 64 2 1∕𝑎 1
= 2 1∕𝑎−2∕𝑎 = 2 −1∕𝑎 =
2 2∕𝑎 2 1∕𝑎
40.
41. 3 2∕33 −1∕6 = 3 2∕3−1∕6 = 3 1∕2 = √3 42. 2 1∕32 −12 2∕32 −1∕3 = 2 1∕3−1+2∕3−1∕3
1
= 2 −1∕3 =
2 1∕3
Edition Stefan Waner
Notes
1- The file is chapter after chapter.
2- We have shown you 10 pages.
3- The file contains all Appendix and Excel
sheet if it exists.
4- We have all what you need, we make
update at every time. There are many
new editions waiting you.
5- If you think you purchased the wrong file
You can contact us at every time, we can
replace it with true one.
Our email:
, Solutions Section 0.1
Section 0.1
1. 2(4 + (−1))(2 ⋅ −4) 2. 3 + ([4 − 2] ⋅ 9)
= 2(3)(−8) = (6)(−8) = −48 = 3 + (2 × 9) = 3 + 18 = 21
3. 20∕(3 * 4) − 1 4. 2 − (3 * 4)∕10
= 20
12
− 1 = 53 − 1 = 2
3
= 2 − 1210
= 2 − 65 = 4
5
3 + ([3 + (−5)]) 12 − (1 − 4)
3−2×2 2(5 − 1) ⋅ 2 − 1
5. 6.
3 + (−2) 1 12 − (−3) 15
= = = −1 = = =1
3−4 −1 16 − 1 15
7. (2 − 5 * (−1))∕1 − 2 * (−1) 8. 2 − 5 * (−1)∕(1 − 2 * (−1))
2 − 5 ⋅ (−1) 5 ⋅ (−1)
= − 2 ⋅ (−1) =2−
1 1 − 2 ⋅ (−1)
2+5 −5 5 11
= +2=7+2=9 =2− +2=2+ =
1 1+2 3 5
2 × (−1) 2 10. 2 + 4 ⋅ 3 2 = 2 + 4 × 9 = 2 + 36 = 38
9. 2 ⋅ (−1) 2∕2 =
2
2×1 2
= = =1
2 2
11. 2 ⋅ 4 2 + 1 12. 1 − 3 ⋅ (−2) 2 × 2
= 2 × 16 + 1 = 32 + 1 = 33 = 1 − 3 × 4 × 2 = 1 − 24 = −23
= 3 2 + 2 2 + 1 = 9 + 4 + 1 = 14 = 2 (2
2−2)
= 2 4−2 = 2 2 = 4
13. 3^2+2^2+1 14. 2^(2^2-2)
3 − 2(−3) 2 1 − 2(1 − 4) 2
−6(4 − 1) 2 2(5 − 1) 2 ⋅ 2
15. 16.
3 − 2 × 9 3 − 18 1 − 2(−3) 2 1−2×9
= = = =
−6(3) 2 −6 × 9 2(4) 2 ⋅ 2 2 × 16 × 2
−15 5 1 − 18 17
= = = =−
−54 18 64 64
1 3 121 121
= =
17. 10*(1+1/10)^3 18. 121/(1+1/10)^2
= 10(1 + ) = 10(1.1) 3
10 !1 + "
1 2 1.1 2
= 10 × 1.331 = 13.31 10
121
= = 100
1.21
, Solutions Section 0.1
−2 ⋅ 3 2 8(1 − 4) 2
19. 3 20. −
[ −(4 − 1) 2 ] [ −9(5 − 1) 2 ]
−2 × 9 ⎤ −18 8×9 72
= 3⎡ = 3[ = −[ = −(
⎣⎢ −3 ⎦⎥
2 −9 ] −9 × 16 ] −144 )
=3×2=6 1 1
= −(− ) =
2 2
1 2 2 1 2 2 2
21. 3⎡1 − (− ) ⎤ + 1 22. 3⎡ − ( ) ⎤ + 1
⎣⎢ 2 ⎦⎥ ⎣⎢ 9 3 ⎦⎥
1 2 1 4 2
= 3[1 − ] + 1 = 3[ − ] + 1
4 9 9
3 2 9 −3 2 −1 2
= 3[ ] + 1 = 3[ ] + 1 = 3[ ] + 1 = 3[ ] + 1
4 16 9 3
27 43 1 3 4
= +1= =3 +1= +1=
16 16 9 9 3
1 2 2
= #2$ − 2 = − =0 = − # 21 $ = − 41 = −2
23. (1/2)^2-1/2^2 24. 2/(1^2)-(2/1)^2
1 1 1 2 2
2 4 4 12 1
25. 3 × (2 − 5) = 3*(2-5) 5
26. 4 +
9
= 4+5/9 or 4+(5/9)
3 4−1
2−5 3
27. = 3/(2-5) 28. = (4-1)/3
3−1 3
30. 3 +
8+6 2−9
29. = (3-1)/(8+6) = 3+3/(2-9)
3 − 18 − 6.
Note 3-1/8-6 is wrong, as it corresponds to
4+7 4×2
31. 3 −
8 2
= 3-(4+7)/8 32. = 4*2/(2/3) or (4*2)/(2/3)
!3"
2 3+𝑥
− 𝑥𝑦 2 = 2/(3+x)-x*y^2 34. 3 +
3+𝑥 𝑥𝑦
33. = 3+(3+x)/(x*y)
60 𝑥2 − 3
35. 3.1𝑥 3 − 4𝑥 −2 − 36. 2.1𝑥 −3 − 𝑥 −1 +
𝑥2 − 1 2
= 3.1x^3-4x^(-2)-60/(x^2-1) = 2.1x^(-3)-x^(-1)+(x^2-3)/2
, 2
Solutions Section 0.1
2
!3"
3
38.
5
37. !5"
= (2/3)/5 = 2/(3/5)
39. 3 4−5 × 6 2
3+5 7−9
40.
= 3^(4-5)*6
= 2/(3+5^(7-9))
Note that the entire exponent is in parentheses.
Note that the entire exponent is in parentheses.
4 1+4
−3 −3
41. 3 1 + 42. 3
( 100 ) ( 100 )
= 3*(1+4/100)^(-3) = 3((1+4)/100)^(-3)
43. 3 2𝑥−1 + 4 𝑥 − 1 2
44. 2 𝑥 − (2 2𝑥) 2
= 3^(2*x-1)+4^x-1 = 2^(x^2)-(2^(2*x))^2
2 2
45. 2 2𝑥 −𝑥+1 46. 2 2𝑥 −𝑥 + 1
= 2^(2x^2-x+1) = 2^(2x^2-x)+1
Note that the entire exponent is in parentheses. Note that the entire exponent is in parentheses.
4𝑒 −2𝑥 𝑒 2𝑥 + 𝑒 −2𝑥
2 − 3𝑒 −2𝑥 𝑒 2𝑥 − 𝑒 −2𝑥
47. 48.
= 4*e^(-2*x)/(2-3e^(-2*x)) = (e^(2*x)+e^(-2*x))/(e^(2*x)-
or 4(*e^(-2*x))/(2-3e^(-2*x)) e^(-2*x))
or (4*e^(-2*x))/(2-3e^(-2*x))
2 2 1 2 2 2
49. 3(1 − !− 12 " ) + 1 50. 3⎛ − ( ) ⎞ + 1
⎜⎝ 9 3 ⎟⎠
= 3(1-(-1/2)^2)^2+1 = 3(1/9-(2/3)^2)^2+1
, Solutions Section 0.2
Section 0.2
1. 3 3 = 27 2. (−2) 3 = −8
3. −(2 ⋅ 3) 2 = −(2 2 ⋅ 3 2) = −(4 ⋅ 9) = −36 4. (4 ⋅ 2) 2 = 4 2 ⋅ 2 2 = 16 ⋅ 4 = 64 or
(4 ⋅ 2) 2 = 8 2 = 64
−(2 ⋅ 3) 2 = −6 2 = −36
or
−2 2 (−2) 2 4 3 3 3 3 27
6. ( ) = 3 =
5. ( = =
3 ) 32 9 2 2 8
1 1 1 1 1
7. (−2) −3 = = =− 8. −2 −3 = − =−
(−2) 3 −8 8 23 8
1 −2 1 1 −2 −2 1 1 9
9. ( ) = = = 16 10. ( = = =
4 (1∕4) 2 1∕16 3 ) (−2∕3) 2 4∕9 4
11. 2 ⋅ 3 0 = 2 ⋅ 1 = 2 12. 3 ⋅ (−2) 0 = 3 ⋅ 1 = 3
13. 2 32 2 = 2 3+2 = 2 5 = 32 14. 3 23 = 3 23 1 = 3 2+1 = 3 3 = 27or
3 23 = 9 ⋅ 3 = 27
2 32 2 = 8 ⋅ 4 = 32
or
15. 2 22 −12 42 −4 = 2 2−1+4−4 = 2 1 = 2 1
16. 5 25 −35 25 −2 = 5 2−3+2−2 = 5 −1 =
5
17. 𝑥 3𝑥 2 = 𝑥 3+2 = 𝑥 5 18. 𝑥 4𝑥 −1 = 𝑥 4−1 = 𝑥 3
𝑦 20. −𝑥𝑦 −1𝑥 −1 = −𝑥 1−1𝑦 −1 = −𝑥 0𝑦 −1
19. −𝑥 2𝑥 −3𝑦 = −𝑥 2−3𝑦 = −𝑥 −1𝑦 = −
𝑥 1
=−
𝑦
𝑥3 1 𝑦5
= 𝑥 3−4 = 𝑥 −1 = = 𝑦 5−3 = 𝑦 2
𝑥4 𝑥 𝑦 3
21. 22.
𝑥 2𝑦 2 𝑥 −1𝑦 1
= 𝑥 2−(−1)𝑦 2−1 = 𝑥 3𝑦 = 𝑥 −1−2𝑦 1−2 = 𝑥 −3𝑦 −1 = 3
𝑥 −1𝑦 𝑥 2𝑦 2 𝑥 𝑦
23. 24.
, (𝑥𝑦 −1𝑧 3) 2 𝑥 2(𝑦 −1) 2(𝑧 3) 2 𝑥 2𝑦𝑧 2 𝑥 2𝑦𝑧 2
Solutions Section 0.2
= =
𝑥 2𝑦𝑧 2 𝑥 2𝑦𝑧 2 (𝑥𝑦𝑧 −1) −1 𝑥 −1𝑦 −1(𝑧 −1) −1
25. 26.
𝑥 𝑦 𝑧
2 −2 6
𝑥 2𝑦𝑧 2
= 2 2 = 𝑥 2−2𝑦 −2−1𝑧 6−2 = −1 −1 = 𝑥 2−(−1)𝑦 1−(−1)𝑧 2−1
𝑥 𝑦𝑧 𝑥 𝑦 𝑧
𝑧4 = 𝑥 3𝑦 2𝑧 4
= 𝑥 0𝑦 −3𝑧 4 = 3
𝑦
𝑥𝑦 −2𝑧 3 (𝑥𝑦 −2𝑧) 3 𝑥 2𝑦 −1𝑧 0 (𝑥 2𝑦 −1𝑧 0) 2
2
" = 28. ( =
𝑥 −1𝑧 (𝑥 −1𝑧) 3 𝑥𝑦𝑧 ) (𝑥𝑦𝑧) 2
27. !
𝑥 𝑦 𝑧
3 −6 3
𝑥 4𝑦 −2
= −3 3 = 𝑥 3−(−3)𝑦 −6𝑧 3−3 = 2 2 2 = 𝑥 4−2𝑦 −2−2𝑧 −2
𝑥 𝑧 𝑥 𝑦 𝑧
𝑥6 𝑥2
= 𝑥 6𝑦 −6𝑧 0 = = 𝑥 2𝑦 −4𝑧 −2 = 4 2
𝑦6 𝑦 𝑧
𝑥 −1𝑦 −2𝑧 2 𝑥𝑦 −2
−2 −3
29. ( ) = (𝑥 𝑦 𝑧 )
−1−1 −2−1 2 −2
= (𝑥 1−2𝑦 −2+1𝑧 −1) −3
𝑥𝑦 ( 𝑥 2𝑦 −1𝑧 )
30.
= (𝑥 −2𝑦 −3𝑧 2) −2 = 𝑥 4𝑦 6𝑧 −4 = (𝑥 −1𝑦 −1𝑧 −1) −3 = 𝑥 3𝑦 3𝑧 3
𝑥 4𝑦 6
= 4
𝑧
31. √4 = 2 32. √5 ≈ 2.236
1 √1 1 1 √1 1
= = = =
√4 √4 2 √9 √9 3
33. 34.
16 √16 4 9 √9 3
= = = =
√9 √9 3 √4 √4 2
35. 36.
√4 2 6 6
= =
5 5 √25 5
37. 38.
39. √9 + √16 = 3 + 4 = 7 40. √25 − √16 = 5 − 4 = 1
41. √9 + 16 = √25 = 5 42. √25 − 16 = √9 = 3
3 3 4 4
43. √8 − 27 = √−19 ≈ −2.668 44. √81 − 16 = √65 ≈ 2.839
3 3 3 3
3 √27 3 √8 × 64 = √8 ⋅ √64 =2⋅4=8
45. √27∕8 = =
2
46.
3
√8
, Solutions Section 0.2
47. √(−2) 2 = √4 = 2 48. √(−1) 2 = √1 = 1
1 16 √16 4 1 36 √36 6
(1 + 15) = = = =2 (3 + 33) = = = =2
√4 √4 √4 2 √9 √9 √9 3
49. 50.
51. √𝑎 2𝑏 2 = √𝑎 2√𝑏 2 = 𝑎𝑏 𝑎 2 √𝑎 2 𝑎
52. √ = =
𝑏 2 √𝑏 2 𝑏
53. √(𝑥 + 9) 2 = 𝑥 + 9
2
54. (√𝑥 + 9) = 𝑥 + 9
(𝑥 + 9 > 0 because 𝑥 is positive.)
3 3 3 4 𝑥4 4
55. √𝑥 3(𝑎 3 + 𝑏 3) = √𝑥 3 √(𝑎 3 + 𝑏 3) √𝑥 4 𝑥
56. √ = =
3
= 𝑥 √(𝑎 3 + 𝑏 3) (Not 𝑥(𝑎 + 𝑏)) 𝑎 4𝑏 4 4 4
√𝑎 4 √𝑏 4 𝑎𝑏
4𝑥𝑦 3 4𝑦 2 √4√𝑦 2 2𝑦 4(𝑥 2 + 𝑦 2) √ 4√ 𝑥 2 + 𝑦2
= √ 𝑥 = = =
√ 𝑥 2𝑦 √𝑥 √𝑥 √ 𝑐2 √𝑐 2
57. 58.
2√𝑥 2 + 𝑦 2
= (Not 2(𝑥 + 𝑦)∕𝑐)
𝑐
, Solutions Section 0.3
Section 0.3
3 1 −4 1
1. 3𝑥 −4 = 𝑥 = 4
𝑥4 2 2𝑥
2.
3 −2∕3 3 4 −3∕4 4
𝑥 = 𝑦 =
4 4𝑥 2∕3 5 5𝑦 3∕4
3. 4.
5. 1 − 0.3
− 65 𝑥 −1 = 1 − 0.3𝑥 2 − 5𝑥
6
1 0.1𝑥 −2 𝑥 4 0.1
𝑥 −2 + = + 2
3𝑥 −4 3 3 3𝑥
6.
3 5
7. 2 2∕3 = √2 2 8. 3 4∕5 = √3 4
3 3 4 4 4 4
10. 𝑦 7∕4 = √𝑦 7 = √𝑦 4 √𝑦 3 = |𝑦| √𝑦 3
3 3
9. 𝑥 4∕3 = √𝑥 4 = √𝑥 3 √𝑥 = 𝑥 √𝑥
11. (𝑥 1∕2𝑦 1∕3)
1∕5 5
= √√ 𝑥 √ 𝑦
3 𝑦 3∕2 √𝑦 3 |𝑦|√𝑦
12. 𝑥 −1∕3𝑦 3∕2 = = 3 = 3
𝑥 1∕3 √𝑥 √𝑥
3 3 3 4 3∕2 4√𝑥 3 4|𝑥|√𝑥
13. − 𝑥 −1∕4 = − =− 4 𝑥 = =
2 2𝑥 1∕4
2 √𝑥
14.
5 5 5
3 0.2 3𝑥 1∕2 3.1 11 −1∕7 11
15. 0.2𝑥 −2∕3 + = + − 𝑥 = 3.1𝑥 4∕3 −
7𝑥 −1∕2 𝑥 2∕3 7 𝑥 −4∕3 7 7𝑥 1∕7
16.
0.2 3√ 𝑥
3 11 3 11
= 3 + = 3.1 √𝑥 4 − 7 = 3.1𝑥 √𝑥 − 7
√𝑥 2 7 7 √𝑥 7 √𝑥
3 3 9 9(1 − 𝑥) 7∕3
= =
4(1 − 𝑥) 5∕2
4√(1 − 𝑥) 5 4(1 − 𝑥) −7∕3 4
17. 18.
3 3 3
9 √(1 − 𝑥) 7 9 √(1 − 𝑥) 6 √1 − 𝑥
3
=
4√(1 − 𝑥) 4√1 − 𝑥 = =
4 4
3
= 3
9(1 − 𝑥) 2 √1 − 𝑥
4(1 − 𝑥) 2√1 − 𝑥 =
4
19. √3 = 3 1∕2 20. √8 = 8 1∕2
21. √𝑥 3 = 𝑥 3∕2 3
22. √𝑥 2 = 𝑥 2∕3
, Solutions Section 0.3
3
23. √𝑥𝑦 2 = (𝑥𝑦 2) 1∕3 24. √𝑥 2𝑦 = (𝑥 2𝑦) 1∕2
𝑥2 𝑥2 𝑥 𝑥
= = 𝑥 2−1∕2 = 𝑥 3∕2 = = 𝑥 1−1∕2 = 𝑥 1∕2
√𝑥 𝑥 1∕2 √𝑥 𝑥 1∕2
25. 26.
3 3 2 2
= 𝑥 −2 = 𝑥3
5𝑥 2 5 5𝑥 −3 5
27. 28.
3𝑥 −1.2 1 3 1 2 𝑥 2.1 2 1.2 1 2.1
− 2.1 = 𝑥 −1.2 − 𝑥 −2.1 − = 𝑥 − 𝑥
2 3𝑥 2 3 3𝑥 −1.2 3 3 3
29. 30.
2𝑥 𝑥 0.1 4 4𝑥 2 𝑥 3∕2 2
− + 1.1 + − 2
3 2 3𝑥 3 6 3𝑥
31. 32.
2 1 4 4 1 2
= 𝑥 − 𝑥 0.1 + 𝑥 −1.1 = 𝑥 2 + 𝑥 3∕2 − 𝑥 −2
3 2 3 3 6 3
3√𝑥 5 4 3 5√𝑥 7
− + − + 3
4 3√𝑥 3𝑥√𝑥 5√𝑥 8 2 √𝑥
33. 34.
3𝑥 1∕2 5 4 3 5𝑥 1∕2 7
= − + = − +
4 3𝑥 1∕2 3𝑥 ⋅ 𝑥 1∕2 5𝑥 1∕2 8 2𝑥 1∕3
3 5 4 3 −1∕2 5 1∕2 7 −1∕3
= 𝑥 1∕2 − 𝑥 −1∕2 + 𝑥 −3∕2 = 𝑥 − 𝑥 + 𝑥
4 3 3 5 8 2
5 1 2 1 2
3 √𝑥 2 7 3𝑥 2∕5 7 − = −
− = − 8𝑥 𝑥 3 √𝑥 3 8𝑥
5 3∕2 3𝑥 3∕5
4 4 2𝑥 3∕2
√
2√𝑥 3
36.
35.
3 7 1 2
= 𝑥 2∕5 − 𝑥 −3∕2 = 𝑥 −3∕2 − 𝑥 −3∕5
4 2 8 3
1 3 3
3 √(𝑥 2 + 1) 7
− 2
(𝑥 2 + 1) 3 3 −
4 √𝑥 2 + 1 3(𝑥 2 + 1) −3 4
37.
1 3
38.
= − 2 2 3
= (𝑥 + 1) 3 − (𝑥 2 + 1) 7∕3
(𝑥 2 + 1) 3 (𝑥 2 + 1) 1∕3 3 4
3
= (𝑥 2 + 1) −3 − (𝑥 2 + 1) −1∕3
4
39. 4 −1∕24 7∕2 = 4 −1∕2+7∕2 = 4 6∕2 = 4 3 = 64 2 1∕𝑎 1
= 2 1∕𝑎−2∕𝑎 = 2 −1∕𝑎 =
2 2∕𝑎 2 1∕𝑎
40.
41. 3 2∕33 −1∕6 = 3 2∕3−1∕6 = 3 1∕2 = √3 42. 2 1∕32 −12 2∕32 −1∕3 = 2 1∕3−1+2∕3−1∕3
1
= 2 −1∕3 =
2 1∕3
