Stewart Precalc 6e Section 1.4: Rational Expressions

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Stewart Precalc 6e Section 1.4: Rational Expressions 0/113
1 Concept - Identifying Rational Expressions · Level 1
Which of the following are rational expressions?
(a) \(\dfrac{3x}{x^2 - 1}\)
(b) \(\dfrac{\sqrt{x+1}}{2x+3}\)
(c) \(\dfrac{x(x^2-1)}{x+3}\)

Enter your answer directly below each part above.

2 Concept - Simplifying Rational Expressions · Level 1
To simplify a rational expression, we cancel factors that are common to the ___ and ___. So the expression \(\dfrac{(x+1)(x+2)}{(x+3)(x+2)}\) simplifies to ___.
3 Concept - Multiplying Rational Expressions · Level 1
To multiply two rational expressions, we multiply their ___ together and multiply their ___ together. So \(\dfrac{2}{x+1} \cdot \dfrac{x}{x+3}\) is the same as ___.
4 Concept - LCD and Combining Fractions · Level 2
Consider the expression \(\dfrac{1}{x} - \dfrac{2}{x+1} - \dfrac{x}{(x+1)^2}\).
(a) How many terms does this expression have?
(b) Find the least common denominator of all the terms.
(c) Perform the addition and simplify.

Enter your answer directly below each part above.

5 Domain of an Expression · Level 1
Find the domain of the expression \(4x^2 - 10x + 3\).
6 Domain of an Expression · Level 1
Find the domain of the expression \(-x^4 + x^3 + 9x\).
7 Domain of an Expression · Level 2
Find the domain of the expression \(\dfrac{2x+1}{x-4}\).
8 Domain of an Expression · Level 2
Find the domain of the expression \(\dfrac{2t^2 - 5}{3t + 6}\).
9 Domain of an Expression · Level 2
Find the domain of the expression \(\sqrt{x + 3}\).
10 Domain of an Expression · Level 2
Find the domain of the expression \(\dfrac{1}{\sqrt{x - 1}}\).
11 Domain of an Expression · Level 2
Find the domain of the expression \(\dfrac{x^2 + 1}{x^2 - x - 2}\).
12 Domain of an Expression · Level 2
Find the domain of the expression \(\dfrac{\sqrt{2x}}{x + 1}\).
13 Simplify Rational Expression · Level 2
Simplify the rational expression \(\dfrac{3(x+2)(x-1)}{6(x-1)^2}\).
14 Simplify Rational Expression · Level 2
Simplify the rational expression \(\dfrac{4(x^2 - 1)}{12(x+2)(x-1)}\).
15 Simplify Rational Expression · Level 2
Simplify the rational expression \(\dfrac{x - 2}{x^2 - 4}\).
16 Simplify Rational Expression · Level 2
Simplify the rational expression \(\dfrac{x^2 - x - 2}{x^2 - 1}\).
17 Simplify Rational Expression · Level 2
Simplify the rational expression \(\dfrac{x^2 + 6x + 8}{x^2 + 5x + 4}\).
18 Simplify Rational Expression · Level 2
Simplify the rational expression \(\dfrac{x^2 - x - 12}{x^2 + 5x + 6}\).
19 Simplify Rational Expression · Level 2
Simplify the rational expression \(\dfrac{y^2 + y}{y^2 - 1}\).
20 Simplify Rational Expression · Level 3
Simplify the rational expression \(\dfrac{y^2 - 3y - 18}{2y^2 + 5y + 3}\).
21 Simplify Rational Expression · Level 3
Simplify the rational expression \(\dfrac{2x^3 - x^2 - 6x}{2x^2 - 7x + 6}\).
22 Simplify Rational Expression · Level 3
Simplify the rational expression \(\dfrac{1 - x^2}{x^3 - 1}\).
23 Multiply or Divide Rational Expressions · Level 2
Perform the multiplication and simplify \(\dfrac{4x}{x^2 - 4} \cdot \dfrac{x + 2}{16x}\).
24 Multiply or Divide Rational Expressions · Level 2
Perform the multiplication and simplify \(\dfrac{x^2 - 25}{x^2 - 16} \cdot \dfrac{x + 4}{x + 5}\).
25 Multiply or Divide Rational Expressions · Level 3
Perform the multiplication and simplify \(\dfrac{x^2 - 2x - 15}{x^2 - 9} \cdot \dfrac{x + 3}{x - 5}\).
26 Multiply or Divide Rational Expressions · Level 3
Perform the multiplication and simplify \(\dfrac{x^2 + 2x - 3}{x^2 - 2x - 3} \cdot \dfrac{3 - x}{3 + x}\).
27 Multiply or Divide Rational Expressions · Level 2
Perform the multiplication and simplify \(\dfrac{t - 3}{t^2 + 9} \cdot \dfrac{t + 3}{t^2 - 9}\).
28 Multiply or Divide Rational Expressions · Level 3
Perform the multiplication and simplify \(\dfrac{x^2 - x - 6}{x^2 + 2x} \cdot \dfrac{x^3 + x^2}{x^2 - 2x - 3}\).
29 Multiply or Divide Rational Expressions · Level 3
Perform the multiplication and simplify \(\dfrac{x^2 + 7x + 12}{x^2 + 3x + 2} \cdot \dfrac{x^2 + 5x + 6}{x^2 + 6x + 9}\).
30 Multiply or Divide Rational Expressions · Level 4
Perform the multiplication and simplify \(\dfrac{x^2 + 2x y + y^2}{x^2 - y^2} \cdot \dfrac{2x^2 - x y - y^2}{x^2 - x y - 2y^2}\).
31 Multiply or Divide Rational Expressions · Level 3
Perform the division and simplify \(\dfrac{x + 3}{4x^2 - 9} \div \dfrac{x^2 + 7x + 12}{2x^2 + 7x - 15}\).
32 Multiply or Divide Rational Expressions · Level 3
Perform the division and simplify \(\dfrac{2x + 1}{2x^2 + x - 15} \div \dfrac{6x^2 - x - 2}{x + 3}\).
33 Multiply or Divide Rational Expressions · Level 4
Perform the division and simplify \(\dfrac{2x^2 + 3x + 1}{x^2 + 2x - 15} \div \dfrac{x^2 + 6x + 5}{2x^2 - 7x + 3}\).
34 Multiply or Divide Rational Expressions · Level 4
Perform the division and simplify \(\dfrac{4y^2 - 9}{2y^2 + 9y - 18} \div \dfrac{2y^2 + y - 3}{y^2 + 5y - 6}\).
35 Compound Quotient · Level 3
Simplify \(\dfrac{x^3 / (x+1)}{x / (x^2 + 2x + 1)}\).
36 Compound Quotient · Level 4
Simplify \(\dfrac{\dfrac{2x^2 - 3x - 2}{x^2 - 1}}{\dfrac{2x^2 + 5x + 2}{x^2 + x - 2}}\).
37 Compound Quotient · Level 1
Simplify \(\dfrac{\dfrac{x}{y}}{z}\).
38 Compound Quotient · Level 1
Simplify \(\dfrac{x}{\dfrac{y}{z}}\).
39 Add or Subtract Rational Expressions · Level 2
Perform the addition and simplify \(2 + \dfrac{x}{x + 3}\).
40 Add or Subtract Rational Expressions · Level 2
Perform the subtraction and simplify \(\dfrac{2x - 1}{x + 4} - 1\).
41 Add or Subtract Rational Expressions · Level 2
Perform the addition and simplify \(\dfrac{1}{x + 5} + \dfrac{2}{x - 3}\).
42 Add or Subtract Rational Expressions · Level 2
Perform the addition and simplify \(\dfrac{1}{x + 1} + \dfrac{1}{x - 1}\).
43 Add or Subtract Rational Expressions · Level 2
Perform the subtraction and simplify \(\dfrac{1}{x + 1} - \dfrac{1}{x + 2}\).
44 Add or Subtract Rational Expressions · Level 3
Perform the subtraction and simplify \(\dfrac{x}{x - 4} - \dfrac{3}{x + 6}\).
45 Add or Subtract Rational Expressions · Level 3
Perform the addition and simplify \(\dfrac{x}{(x + 1)^2} + \dfrac{2}{x + 1}\).
46 Add or Subtract Rational Expressions · Level 3
Perform the subtraction and simplify \(\dfrac{5}{2x - 3} - \dfrac{3}{(2x - 3)^2}\).
47 Add or Subtract Rational Expressions · Level 3
Perform the addition and simplify \(u + 1 + \dfrac{u}{u + 1}\).
48 Add or Subtract Rational Expressions · Level 3
Perform the operations and simplify \(\dfrac{2}{a^2} - \dfrac{3}{a b} + \dfrac{4}{b^2}\).
49 Add or Subtract Rational Expressions · Level 3
Perform the addition and simplify \(\dfrac{1}{x^2} + \dfrac{1}{x^2 + x}\).
50 Add or Subtract Rational Expressions · Level 2
Perform the addition and simplify \(\dfrac{1}{x} + \dfrac{1}{x^2} + \dfrac{1}{x^3}\).
51 Add or Subtract Rational Expressions · Level 3
Perform the subtraction and simplify \(\dfrac{2}{x + 3} - \dfrac{1}{x^2 + 7x + 12}\).
52 Add or Subtract Rational Expressions · Level 3
Perform the addition and simplify \(\dfrac{x}{x^2 - 4} + \dfrac{1}{x - 2}\).
53 Add or Subtract Rational Expressions · Level 3
Perform the addition and simplify \(\dfrac{1}{x + 3} + \dfrac{1}{x^2 - 9}\).
54 Add or Subtract Rational Expressions · Level 4
Perform the subtraction and simplify \(\dfrac{x}{x^2 + x - 2} - \dfrac{2}{x^2 - 5x + 4}\).
55 Add or Subtract Rational Expressions · Level 3
Perform the operations and simplify \(\dfrac{2}{x} + \dfrac{3}{x - 1} - \dfrac{4}{x^2 - x}\).
56 Add or Subtract Rational Expressions · Level 4
Perform the operations and simplify \(\dfrac{x}{x^2 - x - 6} - \dfrac{1}{x + 2} - \dfrac{2}{x - 3}\).
57 Add or Subtract Rational Expressions · Level 4
Perform the subtraction and simplify \(\dfrac{1}{x^2 + 3x + 2} - \dfrac{1}{x^2 - 2x - 3}\).
58 Add or Subtract Rational Expressions · Level 4
Perform the operations and simplify \(\dfrac{1}{x + 1} - \dfrac{2}{(x + 1)^2} + \dfrac{3}{x^2 - 1}\).
59 Compound Fractional Expression · Level 3
Simplify the compound fractional expression \(\dfrac{x + \dfrac{1}{x + 2}}{x - \dfrac{1}{x + 2}}\).
60 Compound Fractional Expression · Level 3
Simplify the compound fractional expression \(\dfrac{1 + \dfrac{1}{c - 1}}{1 - \dfrac{1}{c - 1}}\).
61 Compound Fractional Expression · Level 4
Simplify the compound fractional expression \(\dfrac{\dfrac{x + 2}{x - 1} - \dfrac{x - 3}{x - 2}}{\dfrac{x + 2}{x + 2}}\).
62 Compound Fractional Expression · Level 3
Simplify \(\dfrac{x - 3}{x - 4} - \dfrac{x + 2}{x + 1}\).
63 Compound Fractional Expression · Level 4
Simplify the compound fractional expression \(\dfrac{\dfrac{x}{y} - \dfrac{y}{x}}{\dfrac{1}{x^2} - \dfrac{1}{y^2}}\).
64 Compound Fractional Expression · Level 4
Simplify the compound fractional expression \(x - \dfrac{y}{\dfrac{x}{y} + \dfrac{y}{x}}\).
65 Compound Fractional Expression with Negative Exponents · Level 4
Simplify the compound fractional expression \(\dfrac{x^{-2} - y^{-2}}{x^{-1} + y^{-1}}\).
66 Compound Fractional Expression with Negative Exponents · Level 3
Simplify the compound fractional expression \(\dfrac{x^{-1} + y^{-1}}{(x + y)^{-1}}\).
67 Nested Fractional Expression · Level 3
Simplify \(1 - \dfrac{1}{1 - \dfrac{1}{x}}\).
68 Nested Fractional Expression · Level 4
Simplify \(1 + \dfrac{1}{1 + \dfrac{1}{1 + x}}\).
69 Difference Quotient · Level 4
Simplify the fractional expression \(\dfrac{\dfrac{1}{1 + x + h} - \dfrac{1}{1 + x}}{h}\).
70 Difference Quotient with Radicals · Level 4
Simplify the fractional expression \(\dfrac{\dfrac{1}{\sqrt{x + h}} - \dfrac{1}{\sqrt{x}}}{h}\).
71 Difference Quotient · Level 4
Simplify the fractional expression \(\dfrac{\dfrac{1}{(x + h)^2} - \dfrac{1}{x^2}}{h}\).
72 Difference Quotient · Level 3
Simplify the fractional expression \(\dfrac{(x + h)^3 - 7(x + h) - (x^3 - 7x)}{h}\).
73 Simplify Radical Expression · Level 3
Simplify \(\sqrt{1 + \left(\dfrac{x}{\sqrt{1 - x^2}}\right)^2}\).
74 Simplify Radical Expression · Level 4
Simplify \(\sqrt{1 + \left(x^3 - \dfrac{1}{4 x^3}\right)^2}\).
75 Quotient Rule Simplification · Level 4
Simplify the expression \(\dfrac{3(x + 2)^2 (x - 3)^2 - (x + 2)^3 (2)(x - 3)}{(x - 3)^4}\).
76 Quotient Rule Simplification · Level 4
Simplify the expression \(\dfrac{2 x (x + 6)^4 - x^2 (4)(x + 6)^3}{(x + 6)^8}\).
77 Simplify Expression with Fractional Exponents · Level 4
Simplify the expression \(\dfrac{(1 - x^2)^{\dfrac{1}{2}} + x^2 (1 - x^2)^{-\dfrac{1}{2}}}{1 - x^2}\).
78 Simplify Expression with Fractional Exponents · Level 4
Simplify the expression \(\dfrac{3 (1 + x)^{\dfrac{1}{3}} - x (1 + x)^{-\dfrac{2}{3}}}{(1 + x)^{\dfrac{2}{3}}}\).
79 Simplify Expression with Fractional Exponents · Level 4
Simplify the expression \(\dfrac{(7 - 3x)^{\dfrac{1}{2}} + \dfrac{3}{2} x (7 - 3x)^{-\dfrac{1}{2}}}{7 - 3x}\).
80 Rationalize the Denominator · Level 2
Rationalize the denominator of \(\dfrac{1}{2 - \sqrt{3}}\).
81 Rationalize the Denominator · Level 2
Rationalize the denominator of \(\dfrac{2}{3 - \sqrt{5}}\).
82 Rationalize the Denominator · Level 2
Rationalize the denominator of \(\dfrac{2}{\sqrt{2} + \sqrt{7}}\).
83 Rationalize the Denominator · Level 2
Rationalize the denominator of \(\dfrac{1}{\sqrt{x} + 1}\).
84 Rationalize the Denominator · Level 2
Rationalize the denominator of \(\dfrac{y}{\sqrt{3} + \sqrt{y}}\).
85 Rationalize the Denominator · Level 2
Rationalize the denominator of \(\dfrac{2(x - y)}{\sqrt{x} - \sqrt{y}}\).
86 Rationalize the Numerator · Level 2
Rationalize the numerator of \(\dfrac{1 - \sqrt{5}}{3}\).
87 Rationalize the Numerator · Level 2
Rationalize the numerator of \(\dfrac{\sqrt{3} + \sqrt{5}}{2}\).
88 Rationalize the Numerator · Level 2
Rationalize the numerator of \(\dfrac{\sqrt{r} + \sqrt{2}}{5}\).
89 Rationalize the Numerator · Level 3
Rationalize the numerator of \(\dfrac{\sqrt{x} - \sqrt{x + h}}{h \sqrt{x} \sqrt{x + h}}\).
90 Rationalize the Numerator · Level 3
Rationalize the numerator of \(\sqrt{x^2 + 1} - x\).
91 Rationalize the Numerator · Level 3
Rationalize the numerator of \(\sqrt{x + 1} - \sqrt{x}\).
92 True or False Equation · Level 2
State whether the given equation is true for all values of the variables: \(\dfrac{16 + a}{16} = 1 + \dfrac{a}{16}\).
93 True or False Equation · Level 2
State whether the given equation is true for all values of the variables: \(\dfrac{b}{b - c} = 1 - \dfrac{b}{c}\).
94 True or False Equation · Level 2
State whether the given equation is true for all values of the variables: \(\dfrac{2}{4 + x} = \dfrac{1}{2} + \dfrac{2}{x}\).
95 True or False Equation · Level 2
State whether the given equation is true for all values of the variables: \(\dfrac{x + 1}{y + 1} = \dfrac{x}{y}\).
96 True or False Equation · Level 2
State whether the given equation is true for all values of the variables: \(\dfrac{x}{x + y} = \dfrac{1}{1 + y}\).
97 True or False Equation · Level 2
State whether the given equation is true for all values of the variables: \(2 \left(\dfrac{a}{b}\right) = \dfrac{2 a}{2 b}\).
98 True or False Equation · Level 1
State whether the given equation is true for all values of the variables: \(\dfrac{-a}{b} = -\dfrac{a}{b}\).
99 True or False Equation · Level 2
State whether the given equation is true for all values of the variables: \(\dfrac{1 + x + x^2}{x} = \dfrac{1}{x} + 1 + x\).
100 Application - Electrical Resistance · Level 3
Electrical Resistance: If two electrical resistors with resistances \(R_1\) and \(R_2\) are connected in parallel (see the figure), then the total resistance \(R\) is given by \(R = \dfrac{1}{\dfrac{1}{R_1} + \dfrac{1}{R_2}}\).
question image
(a) Simplify the expression for \(R\).
(b) If \(R_1 = 10\) ohms and \(R_2 = 20\) ohms, what is the total resistance \(R\)?

Enter your answer directly below each part above.

101 Discovery - Limiting Behavior · Level 3
Limiting Behavior of a Rational Expression: The rational expression \(\dfrac{x^2 - 9}{x - 3}\) is not defined for \(x = 3\). Complete the tables and determine what value the expression approaches as \(x\) gets closer and closer to \(3\). Why is this reasonable? Factor the numerator of the expression and simplify to see why.
102 Discovery - Discussion of Rationalization · Level 2
Is This Rationalization? In the expression \(\dfrac{2}{\sqrt{x}}\) we would eliminate the radical if we were to square both numerator and denominator. Is this the same thing as rationalizing the denominator?
103 Discovery - Algebraic Errors and Counterexamples · Level 3
Algebraic Errors: For each algebraic error in the table, give a counterexample using numbers that show the formula is not valid. - \(\dfrac{1}{a} + \dfrac{1}{b} \neq \dfrac{1}{a + b}\) - \((a + b)^2 = a^2 + b^2\) - \(\sqrt{a^2 + b^2} = a + b\) - \(\dfrac{a + b}{a} \neq b\) - \((a^3 + b^3)^{\dfrac{1}{3}} = a + b\) - \(a^m / a^n \neq a^{\dfrac{m}{n}}\) - \(a^{-\dfrac{1}{n}} \neq \dfrac{1}{a^n}\)
104 Discovery - Form of an Algebraic Expression · Level 3
The Form of an Algebraic Expression: With appropriate choices for \(A\) and \(B\), an expression can take the form \(A + B\), \(A B\), \(\dfrac{A}{B}\), or \(A^{\dfrac{1}{2}}\). Find the form of the following algebraic expressions.
(a) \(x + \sqrt{1 + \dfrac{1}{x}}\)
(b) \((1 + x^2)(1 + x)^3\)
(c) \(\sqrt[3]{x^4 (4 x^2 + 1)}\)
(d) \(\dfrac{1 - 2 \sqrt{1 + x}}{1 + \sqrt{1 + x^2}}\)

Enter your answer directly below each part above.

105 Example - Simplifying Rational Expressions by Cancellation · Level 2
Simplify: \(\dfrac{x^2 - 1}{x^2 + x - 2}\)
106 Example - Multiplying Rational Expressions · Level 2
Perform the indicated multiplication and simplify: \(\dfrac{x^2 + 2x - 3}{x^2 + 8x + 16} \cdot \dfrac{3x + 12}{x - 1}\)
107 Example - Dividing Rational Expressions · Level 2
Perform the indicated division and simplify: \(\dfrac{x - 4}{x^2 - 4} \div \dfrac{x^2 - 3x - 4}{x^2 + 5x + 6}\)
108 Example - Adding and Subtracting Rational Expressions · Level 3
Perform the indicated operations and simplify. (a) \(\dfrac{3}{x - 1} + \dfrac{x}{x + 2}\). (b) \(\dfrac{1}{x^2 - 1} - \dfrac{2}{(x + 1)^2}\)
109 Example - Simplifying a Compound Fraction · Level 3
Simplify: \(\dfrac{display\left(\dfrac{x}{y}\right) + 1}{1 - display\left(\dfrac{y}{x}\right)}\)
110 Example - Simplifying a Compound Fraction (Calculus) · Level 3
Simplify: \(\dfrac{display\left(\dfrac{1}{a + h}\right) - display\left(\dfrac{1}{a}\right)}{h}\)
111 Example - Simplifying a Compound Fraction (Calculus) · Level 4
Simplify: \(\dfrac{(1 + x^2)^{\dfrac{1}{2}} - x^2 (1 + x^2)^{-\dfrac{1}{2}}}{1 + x^2}\)
112 Example - Rationalizing the Denominator · Level 3
Rationalize the denominator: \(\dfrac{1}{1 + \sqrt{2}}\)
113 Example - Rationalizing the Numerator (Calculus) · Level 3
Rationalize the numerator: \(\dfrac{\sqrt{4 + h} - 2}{h}\)

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