Let’s begin by taking powers of fractions. Recall that
\[ a^m = \underbrace{a \cdot a \cdot ... \cdot a}_{m \text{ times}}\nonumber \]
The last two examples reiterate a principle learned earlier.
Order of Operations
For convenience, we repeat here the rules guiding order of operations.
Rules Guiding Order of Operations
When evaluating expressions, proceed in the following order.
Evaluate expressions contained in grouping symbols first. If grouping symbols are nested, evaluate the expression in the innermost pair of grouping symbols first.
Evaluate all exponents that appear in the expression.
Perform all multiplications and divisions in the order that they appear in the expression, moving left to right.
Perform all additions and subtractions in the order that they appear in the expression, moving left to right.
Example 3
Simplify: \(- \frac{1}{2} + \frac{1}{4} \left( - \frac{1}{3} \right)\).
Solution
Multiply first, then add.
\[ \begin{aligned} - \frac{1}{2} + \frac{1}{4} \left( - \frac{1}{3} \right) = - \frac{1}{2} + \left( - \frac{1}{12} \right) ~ & \textcolor{red}{ \text{ Multiply: } \frac{1}{4} \left( - \frac{1}{3} \right) = - \frac{1}{12}.} \\ = - \frac{1 \cdot \textcolor{red}{6}}{2 \cdot \textcolor{red}{6}} + \left( - \frac{1}{12} \right) ~ & \textcolor{red}{ \text{ Equivalent fractions, LCD = 12.}} \\ = - \frac{6}{12} + \left( - \frac{1}{12} \right) ~ & \textcolor{red}{ \text{ Simplify numerator and denominator.}} \\ = - \frac{7}{12} ~ & \textcolor{red}{ \text{ Add over common denominator.}} \end{aligned}\nonumber \]
Exercise
Simplify: \( - \frac{2}{3} + \frac{3}{4} \left( - \frac{1}{2} \right)\)
Answer
−25/24
Example 4
Simplify: \(2 \left( - \frac{1}{2} \right)^2 +4 \left( - \frac{1}{2} \right)\).
Solution
Exponents first, then multiply, then add.
\[ \begin{aligned} 2 \left( - \frac{1}{2} \right)^2 + 4 \left( - \frac{1}{2} \right) = 2 \left( \frac{1}{4} \right) + 4 \left( - \frac{1}{2} \right) ~ & \textcolor{red}{ \text{ Exponent first: } \left( - \frac{1}{2} \right)^2 = \frac{1}{4}.} \\ = \frac{1}{2} + \left( - \frac{2}{1} \right) ~ & \textcolor{red}{ \begin{array}{l} \text{ Multiply: } 2 \left( \frac{1}{4} \right) = \frac{1}{2} \\ \text{ and } 4 \left( - \frac{1}{2} \right) = - \frac{2}{1}. \end{array}} \\ = \frac{1}{2} + \left( - \frac{2 \cdot \textcolor{red}{2}}{1 \cdot \textcolor{red}{2}} \right) ~ & \textcolor{red}{ \text{ Equivalent fractions, LCD = 2.}} \\ = \frac{1}{2} + \left( - \frac{4}{2} \right) ~ & \textcolor{red}{ \text{ Simplify numerator and denominator.}} \\ = - \frac{3}{2} ~ & \textcolor{red}{ \text{ Add over common denominator.}} \end{aligned}\nonumber \]
Exercise
Simplify: \(3 \left( - \frac{1}{3} \right)^2 - 2 \left( - \frac{1}{3} \right)\)
Answer
1
Example 5
Given a = −3/4, b = 1/2, c = 1/3, and d = −1/4, evaluate the expression ab − cd .
Solution
Recall that it is good practice to prepare parentheses before substituting.
\[ ad - bc = ( ~ ) (~) - (~)(~)\nonumber \]
Substitute the given values into the algebraic expression, then simplify using order of operations.
\[ \begin{aligned} ab - cd = \left( - \frac{3}{4} \right) \left( \frac{1}{2} \right) - \left( \frac{1}{3} \right) \left( - \frac{1}{4} \right) ~ & \textcolor{red}{ \begin{array}{l} \text{ Substitute: } -3/4 \text{ for } a, ~ 1/2 \text{ for } b, \\ 1/3 \text{ for } c, ~ \text{ and } -1/4 \text{ for } d. \end{array}} \\ = - \frac{3}{8} - \left( - \frac{1}{12} \right) ~ & \textcolor{red}{ \begin{array}{l} \text{ Multiply first: } \left( - \frac{3}{4} \right) \left( \frac{1}{2} \right) = - \frac{3}{8} \\ \text{ and } \left( \frac{1}{3} \right) \left( - \frac{1}{4} \right) = - \frac{1}{12}. \end{array}} \\ = - \frac{3}{8} + \frac{1}{12} ~ & \textcolor{red}{ \text{ Subtract by adding opposite.}} \\ = - \frac{3 \cdot \textcolor{red}{3}}{8 \cdot \textcolor{red}{3}} + \frac{1 \cdot \textcolor{red}{2}}{12 \cdot \textcolor{red}{2}} ~ & \textcolor{red}{ \text{ Equivalent fractions; LCD = 24.}} \\ = - \frac{9}{24} + \frac{2}{24} ~ & \textcolor{red}{ \text{ Simplify numerators and denominators.}} \\ = - \frac{7}{24} ~ & \textcolor{red}{ \text{ Add over common denominator.}} \end{aligned}\nonumber \]
Exercise
Given a = −1/2, b = 1/3, and c = −1/5, evaluate a + bc .
Answer
−17/30
Example 6
Given a = −1/4 and b = 1/2, evaluate (a 2 − b 2 ) ÷ (a + b ).
Solution
Recall that it is good practice to prepare parentheses before substituting.
\[ (a^2 - b^2 ) \div (a+b) - \left( (~)^2 - (~)^2 \right) \div \left( (~) + (~) \right)\nonumber \]
Substitute the given values into the algebraic expression, then evaluate exponents first.
\[ \begin{array}{l} (a^2 -b^2) \div (a+b) & = \left( \left( - \frac{1}{4} \right)^2 - \left( \frac{1}{2} \right)^2 \right) \div \left( \left( - \frac{1}{4} \right) + \left( \frac{1}{2} \right) \right) \\ ~ & = \left( \frac{1}{16} - \frac{1}{4} \right) \div \left( - \frac{1}{4} + \frac{1}{2} \right) \end{array}\nonumber \]
We must evaluate parentheses first. Inside each set of parentheses, create equivalent fractions and perform subtractions and additions next.
\[ \begin{array}{l} = \left( \frac{1}{16} - \frac{1 \cdot 4}{4 \cdot 4} \right) \div \left( - \frac{1}{4} + \frac{1 \cdot 2}{2 \cdot 2} \right) \\ = \left( \frac{1}{16} - \frac{4}{16} \right) \div \left( - \frac{1}{4} + \frac{2}{4} \right) \\ = - \frac{3}{16} \div \frac{1}{4} \end{array}\nonumber \]
Invert and multiply.
\[ \begin{aligned} = - \frac{3}{16} \cdot \frac{4}{1} \\ = - \frac{12}{16} \end{aligned}\nonumber \]
Reduce.
\[ \begin{aligned} = - \frac{12 \div 4}{16 \div 4} \\ - \frac{3}{4} \end{aligned}\nonumber \]
Note: In the last step, you could also reduce by prime factoring numerator and denominator and canceling common factors.
Exercise
Give a = −1/2 and b = −1/3, evaluate ab ÷ (a + b ).
Answer
−1/5
Complex Fractions
Complex Fractions
When the numerator and denominator of a fraction contain fractions themselves, such an expression is called a complex fraction .
You can use the standard order of operations to simplify a complex fraction. Recall the advice when a fraction is present.
Fractional Expressions
If a fractional expression is present, simplify the numerator and denominator separately, then divide.
Example 7
Simplify:
\[ \frac{ - \frac{1}{2} + \frac{1}{3}}{ \frac{3}{4} - \frac{3}{2}}\nonumber \]
Solution
We have addition in the numerator, subtraction in the denominator. In each case, we need equivalent fractions with a common denominator.
\[ \begin{aligned} \frac{- \frac{1}{2} + \frac{1}{3}}{ \frac{3}{4} - \frac{3}{2}} = \frac{- \frac{1 \cdot \textcolor{red}{3}}{2 \cdot \textcolor{red}{3}} + \frac{1 \cdot \textcolor{red}{2}}{3 \cdot \textcolor{red}{2}}} ~ & \textcolor{red}{ \text{ Create equivalent fractions.}} \\ = \frac{- \frac{3}{6} + \frac{2}{6}}{ \frac{3}{4} - \frac{6}{4}} ~ & \textcolor{red}{ \text{ Simplify numerator and denominator.}} \\ = \frac{- \frac{1}{6}}{- \frac{3}{4}} ~ & \textcolor{red}{ \begin{array}{l} ~ \text{ Numerator: } - \frac{3}{6} + \frac{2}{6} = - \frac{1}{6}. \\ \text{ Denominator: } \frac{3}{4} - \frac{6}{4} = - \frac{3}{4}. \end{array}} \end{aligned}\nonumber \]
The last expression asks us to divide. Invert and multiply.
\[ \begin{aligned} = - \frac{1}{6} \div \left( - \frac{3}{4} \right) ~ & \textcolor{red}{ \text{ A complex fraction means divide.}} \\ = - \frac{1}{6} \cdot \left( - \frac{4}{3} \right) ~ & \textcolor{red}{ \text{ Invert and multiply.}} \end{aligned}\nonumber \]
Like signs (two negatives) give a positive product. Multiply numerators and denominators, then reduce.
\[ \begin{aligned} = \frac{4}{18} ~ & \textcolor{red}{ \begin{array}{l} \text{ Like signs yields positive answer.} \\ \text{ Multiply numerators and denominators.} \end{array}} \\ = \frac{4 \div 2}{18 \div 2} ~ & \textcolor{red}{ \text{ Divide both numerator and denominator by 2.}} \\ = \frac{2}{9} ~ & \textcolor{red}{ \text{ Simplify.}} \end{aligned}\nonumber \]
Alternatively, one could prime factor and cancel to reduce to lowest terms; that is,
\[ \begin{aligned} \frac{4}{18} = \frac{2 \cdot 2}{2\cdot 3 \cdot 3} ~ & \textcolor{red}{ \text{ Prime factor.}} \\ = \frac{ \cancel{2} \cdot 2}{ \cancel{2} \cdot 3 \cdot 3} ~ & \textcolor{red}{ \text{ Cancel common factors.}} \\ = \frac{2}{9} ~ & \textcolor{red}{ \text{ Simplify.}} \end{aligned}\nonumber \]
Exercise
Simplify:
\[ \frac{ \frac{1}{4} - \frac{1}{3}}{ \frac{1}{4} + \frac{1}{3}}\nonumber \]
Answer
−1/7
Application — Trapezoid
A trapezoid is a special type of quadrilateral (four-sided polygon).
Trapezoid
A quadrilateral with one pair of parallel opposite sides is called a trapezoid .
The pair of parallel sides are called the bases of the trapezoid. Their lengths are marked by the variables b 1 and b 2 in the figure above. The distance between the parallel bases is called the height or altitude of the trapezoid. The height is marked by the variable h in the figure above.
Mathematicians use subscripts to create new variables. Thus, b 1 (“b sub 1”) and b 2 (“b sub 2”) are two distinct variables, used in this case to represent the length of the bases of the trapezoid.
By drawing in a diagonal, we can divide the trapezoid into two triangles (see Figure 4.14).
Figure 4.14: Dividing the trapezoid into two triangles.
We can find the area of the trapezoid by summing the areas of the two triangles.
The shaded triangle in Figure 4.14 has base b 1 and height h . Hence, the area of the shaded triangle is (1/2)b 1 h .
The unshaded triangle in Figure 4.14 has base b 2 and height h . Hence, the area of the unshaded triangle is (1/2)b 2 h .
Summing the areas, the area of the trapezoid is
\[ \text{Area of Trapezoid} = \frac{1}{2} b_1h + \frac{1}{2} b_2h.\nonumber \]
We can use the distributive property to factor out a (1/2)h .
Area of a Trapezoid
A trapezoid with bases b 1 and b 2 and height h has area
\[A = \frac{1}{2} h (b_1 + b_2).\nonumber \]
That is, to find the area, sum the bases, multiply by the height, and take one-half of the result.
Example 9
Find the area of the trapezoid pictured below.
Solution
The formula for the area of a trapezoid is
\[A = \frac{1}{2} h (b_1 + b_2)\nonumber \]
Substituting the given bases and height, we get
\[A = \frac{1}{2} (3) \left( 4 \frac{1}{4} + 2 \frac{1}{2} \right).\nonumber \]
Simplify the expression inside the parentheses first. Change mixed fractions to improper fractions, make equivalent fractions with a common denominator, then add.
\[\begin{array}{c} A = \frac{1}{2} (3) \left( \frac{17}{4} + \frac{5}{2} \right) \\ = \frac{1}{2} (3) \left( \frac{17}{4} + \frac{5 \cdot 2}{2 \cdot 2} \right) \\ = \frac{1}{2} (3) \left( \frac{17}{4} + \frac{10}{4} \right) \\ = \frac{1}{2} \left( \frac{3}{1} \right) \left( \frac{27}{4} \right) \end{array}\nonumber \]
Multiply numerators and denominators.
\[ = \frac{81}{8}\nonumber \]
This improper fraction is a perfectly good answer, but let’s change this result to a mixed fraction (81 divided by 8 is 10 with a remainder of 1). Thus, the area of the trapezoid is
\[A = 10 \frac{1}{8} \text{ square inches.}\nonumber \]
Exercise
A trapezoid has bases measuring 6 and 15 feet, respectively. The height of the trapezoid is 5 feet. Find the area of the trapezoid.
Answer
\(52 \frac{1}{2} \text{ square feet}\)
Exercises
In Exercises 1-8, simplify the expression.
1. \( \left( − \frac{7}{3} \right)^3\)
2. \( \left( \frac{1}{2} \right)^3\)
3. \( \left( \frac{5}{3} \right)^4\)
4. \( \left( − \frac{3}{5} \right)^4\)
5. \( \left( \frac{1}{2} \right)^5\)
6. \( \left( \frac{3}{4} \right)^5\)
7. \( \left( \frac{4}{3} \right)^2\)
8. \( \left( − \frac{8}{5} \right)^2\)
9. If a = 7/6, evaluate a 3 .
10. If e = 1/6, evaluate e 3 .
11. If e = −2/3, evaluate −e 2 .
12. If c = −1/5, evaluate −c 2 .
13. If b = −5/9, evaluate b 2 .
14. If c = 5/7, evaluate c 2 .
15. If b = −1/2, evaluate −b 3 .
16. If a = −2/9, evaluate −a 3 .
In Exercises 17-36, simplify the expression.
17. \( \left( − \frac{1}{2} \right) \left( \frac{1}{6} \right) − \left( \frac{7}{8} \right) \left( − \frac{7}{9} \right)\)
18. \( \left( − \frac{3}{4} \right) \left( \frac{1}{2} \right) − \left( \frac{3}{5} \right) \left( \frac{1}{4} \right)\)
19. \( \left( − \frac{9}{8} \right)^2 − \left( − \frac{3}{2} \right) \left( \frac{7}{3} \right)\)
20. \( \left( \frac{3}{2} \right)^2 − \left( \frac{7}{8} \right) \left( − \frac{1}{2} \right)\)
21. \( \left( − \frac{1}{2} \right) \left( − \frac{7}{4} \right) − \left( − \frac{1}{2} \right)^2\)
22. \( \left( \frac{1}{5} \right) \left( − \frac{9}{4} \right) − \left( \frac{7}{4} \right)^2\)
23. \(− \frac{7}{6} − \frac{1}{7} \cdot \frac{7}{9}\)
24. \( − \frac{4}{9} − \frac{8}{5} \cdot \frac{8}{9}\)
25. \( \frac{3}{4} + \frac{9}{7} \left( − \frac{7}{6} \right)\)
26. \( \frac{3}{2} + \frac{1}{4} \left( − \frac{9}{8} \right)\)
27. \( \left( − \frac{1}{3} \right)^2 + \left( \frac{7}{8} \right) \left( − \frac{1}{3} \right)\)
28. \( \left( − \frac{2}{9} \right)^2 + \left( \frac{2}{3} \right) \left( \frac{1}{2} \right)\)
29. \(\frac{5}{9} + \frac{5}{9} \cdot \frac{7}{9}\)
30. \( − \frac{1}{2} + \frac{9}{8} \cdot \frac{1}{3}\)
31. \( \left( − \frac{5}{6} \right) \left( \frac{3}{8} \right) + \left( − \frac{7}{9} \right) \left( − \frac{3}{4} \right)\)
32. \( \left( \frac{7}{4} \right) \left( \frac{6}{5} \right) + \left( − \frac{2}{5} \right) \left( \frac{8}{3} \right)\)
33. \( \frac{4}{3} − \frac{2}{9} \left( − \frac{3}{4} \right)\)
34. \(− \frac{1}{3} − \frac{1}{5} \left( − \frac{4}{3} \right)\)
35. \( \left( − \frac{5}{9} \right) \left( \frac{1}{2} \right) + \left( − \frac{1}{6} \right)^2\)
36. \( \left( \frac{1}{4} \right) \left( \frac{1}{6} \right) + \left( − \frac{5}{6} \right)^2\)
37. Given a = −5/4, b = 1/2, and c = 3/8, evaluate a + bc .
38. Given a = −3/5, b = 1/5, and c = 1/3, evaluate a + bc .
39. Given x = −1/8, y = 5/2, and z = −1/2, evaluate the expression x + yz .
40. Given x = −5/9, y = 1/4, and z = −2/3, evaluate the expression x + yz .
41. Given a = 3/4, b = 5/7, and c = 1/2, evaluate the expression a − bc .
42. Given a = 5/9, b = 2/3, and c = 2/9, evaluate the expression a − bc .
43. Given x = −3/2, y = 1/4, and z = −5/7, evaluate x 2 − yz .
44. Given x = −3/2, y = −1/2, and z = 5/3, evaluate x 2 − yz .
45. Given a = 6/7, b = 2/3, c = −8/9, and d = −6/7, evaluate ab + cd .
46. Given a = 4/9, b = −3/2, c = 7/3, and d = −8/9, evaluate ab + cd .
47. Given w = −1/8, x = −2/7, y = −1/2, and z = 8/7, evaluate wx − yz .
48. Given w = 2/7, x = −9/4, y = −3/4, and z = −9/2, evaluate wx − yz .
49. Given x = 3/8, y = 3/5, and z = −3/2, evaluate xy + z 2 .
50. Given x = −1/2, y = 7/5, and z = −3/2, evaluate xy + z 2 .
51. Given u = 9/7, v = 2/3, and w = −3/7, evaluate uv − w 2 .
52. Given u = 8/7, v = −4/3, and w = 2/3, evaluate uv − w 2 .
53. Given a = 7/8, b = −1/4, and c = −3/2, evaluate a 2 + bc .
54. Given a = −5/8, b = 3/2, and c = −3/2, evaluate a 2 + bc .
55. Given u = 1/3, v = 5/2, and w = −2/9, evaluate the expression u − vw .
56. Given u = −1/2, v = 1/4, and w = −1/4, evaluate the expression u − vw .
In Exercises 57-68, simplify the complex rational expression.
57. \(\frac{ \frac{8}{3} + \frac{7}{6}}{− \frac{9}{2} − \frac{1}{4}}\)
58. \( \frac{ \frac{7}{8} + \frac{1}{9}}{ \frac{8}{9} − \frac{1}{6}}\)
59. \( \frac{ \frac{3}{4} + \frac{4}{3}}{ \frac{1}{9} + \frac{5}{3}}\)
60. \( \frac{− \frac{9}{8} − \frac{6}{5}}{ \frac{7}{4} + \frac{1}{2}}\)
61. \( \frac{ \frac{7}{5} + \frac{5}{2}}{− \frac{1}{4} + \frac{1}{2}}\)
62. \( \frac{ \frac{5}{6} + \frac{2}{3}}{ \frac{3}{5} + \frac{2}{3}}\)
63. \( \frac{− \frac{3}{2} − \frac{2}{3}}{− \frac{7}{4} − \frac{2}{3}}\)
64. \( \frac{ \frac{8}{9} + \frac{3}{4}}{− \frac{2}{3} − \frac{1}{6}}\)
65. \( \frac{− \frac{1}{2} − \frac{4}{7}}{− \frac{5}{7} + \frac{1}{6}}\)
66. \( \frac{− \frac{3}{2} − \frac{5}{8}}{ \frac{3}{4} − \frac{1}{2}}\)
67. \(\frac{− \frac{3}{7} − \frac{1}{3}}{ \frac{1}{3} − \frac{6}{7}}\)
68. \( \frac{− \frac{5}{8} − \frac{6}{5}}{− \frac{5}{4} − \frac{3}{8}}\)
69. A trapezoid has bases measuring \(3 \frac{3}{8}\) and \(5 \frac{1}{2}\) feet, respectively. The height of the trapezoid is 7 feet. Find the area of the trapezoid.
70. A trapezoid has bases measuring \(2 \frac{1}{2}\) and \(6 \frac{7}{8}\) feet, respectively. The height of the trapezoid is 3 feet. Find the area of the trapezoid.
71. A trapezoid has bases measuring \(2 \frac{1}{4}\) and \(7 \frac{3}{8}\) feet, respectively. The height of the trapezoid is 7 feet. Find the area of the trapezoid.
72. A trapezoid has bases measuring \(3 \frac{1}{8}\) and \(6 \frac{1}{2}\) feet, respectively. The height of the trapezoid is 3 feet. Find the area of the trapezoid.
73. A trapezoid has bases measuring \(2 \frac{3}{4}\) and \(6 \frac{5}{8}\) feet, respectively. The height of the trapezoid is 3 feet. Find the area of the trapezoid.
74. A trapezoid has bases measuring \(2 \frac{1}{4}\) and \(7 \frac{1}{8}\) feet, respectively. The height of the trapezoid is 5 feet. Find the area of the trapezoid.