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1.6 Rational Expressions

Reducing algebraic fractions to lowest terms, multiplying and dividing them, finding common denominators to add or subtract, and untangling complex fractions.

Algebra and Trigonometry 2e, 1.6 Rational Expressions

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89 exercises

Simplifying Rational Expressions

Simplify x 2 −9 x 2 +4x+3 . \frac{x^{2} - 9}{x^{2} + 4 x + 3} .

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1.6 Exercise 5

Simplify x−6 x 2 −36 . \frac{x - 6}{x^{2} - 36} .

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Multiplying Rational Expressions

Multiply the rational expressions and show the product in simplest form:

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1.6 Exercise 7

Multiply the rational expressions and show the product in simplest form:

x 2 +11x+30 x 2 +5x+6 ⋅ x 2 +7x+12 x 2 +8x+16 \frac{x^{2} + 11 x + 30}{x^{2} + 5 x + 6} \cdot \frac{x^{2} + 7 x + 12}{x^{2} + 8 x + 16}
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Dividing Rational Expressions

Divide the rational expressions and express the quotient in simplest form:

2 x 2 +x−6 x 2 −1 ÷ x 2 −4 x 2 +2x+1 \frac{2 x^{2} + x - 6}{x^{2} - 1} \div \frac{x^{2} - 4}{x^{2} + 2 x + 1}
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1.6 Exercise 9

Divide the rational expressions and express the quotient in simplest form:

9 x 2 −16 3 x 2 +17x−28 ÷ 3 x 2 −2x−8 x 2 +5x−14 \frac{9 x^{2} - 16}{3 x^{2} + 17 x - 28} \div \frac{3 x^{2} - 2 x - 8}{x^{2} + 5 x - 14}
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Adding Rational Expressions

Add the rational expressions:

5 x + 6 y \frac{5}{x} + \frac{6}{y}
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Subtracting Rational Expressions

Subtract the rational expressions:

6 x 2 +4x+4 − 2 x 2 −4 \frac{6}{x^{2} + 4 x + 4} - \frac{2}{x^{2} −4}
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1.6 Exercise 12

Subtract the rational expressions: 3 x+5 − 1 x−3 . \frac{3}{x + 5} - \frac{1}{x −3} .

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Simplifying Complex Rational Expressions

Simplify: y+ 1 x x y \frac{y + \frac{1}{x}}{\frac{x}{y}} .

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1.6 Exercise 14

Simplify: x y − y x y \frac{\frac{x}{y} - \frac{y}{x}}{y}

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1.6 Exercise 15

How can you use factoring to simplify rational expressions?

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1.6 Exercise 17

Tell whether the following statement is true or false and explain why: You only need to find the LCD when adding or subtracting rational expressions.

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1.6 Exercise 19

Simplify the rational expressions.

y 2 +10y+25 y 2 +11y+30 \frac{y^{2} + 10 y + 25}{y^{2} + 11 y + 30}

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1.6 Exercise 21

Simplify the rational expressions.

9 b 2 +18b+9 3b+3 \frac{9 b^{2} + 18 b + 9}{3 b + 3}

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1.6 Exercise 23

Simplify the rational expressions.

2 x 2 +7x−4 4 x 2 +2x−2 \frac{2 x^{2} + 7 x - 4}{4 x^{2} + 2 x - 2}

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1.6 Exercise 25

Simplify the rational expressions.

a 2 +9a+18 a 2 +3a−18 \frac{a^{2} + 9 a + 18}{a^{2} + 3 a - 18}

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1.6 Exercise 27

Simplify the rational expressions.

12 n 2 −29n−8 28 n 2 −5n−3 \frac{12 n^{2} - 29 n - 8}{28 n^{2} - 5 n - 3}

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1.6 Exercise 29

Multiply the rational expressions and express the product in simplest form.

c 2 +2c−24 c 2 +12c+36 ⋅ c 2 −10c+24 c 2 −8c+16 \frac{c^{2} + 2 c - 24}{c^{2} + 12 c + 36} \cdot \frac{c^{2} - 10 c + 24}{c^{2} - 8 c + 16}

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1.6 Exercise 31

Multiply the rational expressions and express the product in simplest form.

10 h 2 −9h−9 2 h 2 −19h+24 ⋅ h 2 −16h+64 5 h 2 −37h−24 \frac{10 h^{2} - 9 h - 9}{2 h^{2} - 19 h + 24} \cdot \frac{h^{2} - 16 h + 64}{5 h^{2} - 37 h - 24}

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1.6 Exercise 33

Multiply the rational expressions and express the product in simplest form.

2 d 2 +15d+25 4 d 2 −25 ⋅ 2 d 2 −15d+25 25 d 2 −1 \frac{2 d^{2} + 15 d + 25}{4 d^{2} - 25} \cdot \frac{2 d^{2} - 15 d + 25}{25 d^{2} - 1}

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1.6 Exercise 35

Multiply the rational expressions and express the product in simplest form.

t 2 −1 t 2 +4t+3 ⋅ t 2 +2t−15 t 2 −4t+3 \frac{t^{2} - 1}{t^{2} + 4 t + 3} \cdot \frac{t^{2} + 2 t - 15}{t^{2} - 4 t + 3}

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1.6 Exercise 37

Multiply the rational expressions and express the product in simplest form.

36 x 2 −25 6 x 2 +65x+50 ⋅ 3 x 2 +32x+20 18 x 2 +27x+10 \frac{36 x^{2} - 25}{6 x^{2} + 65 x + 50} \cdot \frac{3 x^{2} + 32 x + 20}{18 x^{2} + 27 x + 10}

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1.6 Exercise 39

Divide the rational expressions.

6 p 2 +p−12 8 p 2 +18p+9 ÷ 6 p 2 −11p+4 2 p 2 +11p−6 \frac{6 p^{2} + p - 12}{8 p^{2} + 18 p + 9} \div \frac{6 p^{2} - 11 p + 4}{2 p^{2} + 11 p - 6}

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1.6 Exercise 41

Divide the rational expressions.

18 d 2 +77d−18 27 d 2 −15d+2 ÷ 3 d 2 +29d−44 9 d 2 −15d+4 \frac{18 d^{2} + 77 d - 18}{27 d^{2} - 15 d + 2} \div \frac{3 d^{2} + 29 d - 44}{9 d^{2} - 15 d + 4}

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1.6 Exercise 43

Divide the rational expressions.

144 b 2 −25 72 b 2 −6b−10 ÷ 18 b 2 −21b+5 36 b 2 −18b−10 \frac{144 b^{2} - 25}{72 b^{2} - 6 b - 10} \div \frac{18 b^{2} - 21 b + 5}{36 b^{2} - 18 b - 10}

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1.6 Exercise 45

Divide the rational expressions.

22 y 2 +59y+10 12 y 2 +28y−5 ÷ 11 y 2 +46y+8 24 y 2 −10y+1 \frac{22 y^{2} + 59 y + 10}{12 y^{2} + 28 y - 5} \div \frac{11 y^{2} + 46 y + 8}{24 y^{2} - 10 y + 1}

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1.6 Exercise 47

Add and subtract the rational expressions, and then simplify.

4 x + 10 y \frac{4}{x} + \frac{10}{y}

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1.6 Exercise 49

Add and subtract the rational expressions, and then simplify.

4 a+1 + 5 a−3 \frac{4}{a + 1} + \frac{5}{a - 3}

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1.6 Exercise 51

Add and subtract the rational expressions, and then simplify.

y+3 y−2 + y−3 y+1 \frac{y + 3}{y - 2} + \frac{y - 3}{y + 1}

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1.6 Exercise 53

Add and subtract the rational expressions, and then simplify.

3z z+1 + 2z+5 z−2 \frac{3 z}{z + 1} + \frac{2 z + 5}{z - 2}

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1.6 Exercise 55

Add and subtract the rational expressions, and then simplify.

x x+1 + y y+1 \frac{x}{x + 1} + \frac{y}{y + 1}

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1.6 Exercise 57

Simplify the rational expression.

2 a + 7 b b \frac{\frac{2}{a} + \frac{7}{b}}{b}

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1.6 Exercise 59

Simplify the rational expression.

3 a + b 6 2b 3a \frac{\frac{3}{a} + \frac{b}{6}}{\frac{2 b}{3 a}}

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1.6 Exercise 61

Simplify the rational expression.

a b − b a a+b ab \frac{\frac{a}{b} - \frac{b}{a}}{\frac{a + b}{a b}}

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1.6 Exercise 63

Simplify the rational expression.

2c c+2 + c−1 c+1 2c+1 c+1 \frac{\frac{2 c}{c + 2} + \frac{c - 1}{c + 1}}{\frac{2 c + 1}{c + 1}}

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1.6 Exercise 67

Elroi wants to mulch his garden. His garden is x 2 +18x+81 x^{2} + 18 x + 81 ft2. One bag of mulch covers x 2 −81 x^{2} - 81 ft2. Divide the expressions and simplify to find how many bags of mulch Elroi needs to mulch his garden.

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1.6 Exercise 69

Perform the given operations and simplify.

3 y 2 −10y+3 3 y 2 +5y−2 ⋅ 2 y 2 −3y−20 2 y 2 −y−15 y−4 \frac{\frac{3 y^{2} - 10 y + 3}{3 y^{2} + 5 y - 2} \cdot \frac{2 y^{2} - 3 y - 20}{2 y^{2} - y - 15}}{y - 4}

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1.6 Exercise 71

Perform the given operations and simplify.

x 2 +7x+12 x 2 +x−6 ÷ 3 x 2 +19x+28 8 x 2 −4x−24 ÷ 2 x 2 +x−3 3 x 2 +4x−7 \frac{x^{2} + 7 x + 12}{x^{2} + x - 6} \div \frac{3 x^{2} + 19 x + 28}{8 x^{2} - 4 x - 24} \div \frac{2 x^{2} + x - 3}{3 x^{2} + 4 x - 7}

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1.6 Exercise 72

Perform the given operations.

( 5−3⋅2 ) 2 −6 \left(5 - 3 \cdot 2\right)^{2} - 6

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1.6 Exercise 74

Perform the given operations.

2⋅ 5 2 +6÷2 2 \cdot 5^{2} + 6 \div 2

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1.6 Exercise 76

Solve the equation.

2y+ 4 2 =64 2 y + 4^{2} = 64

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1.6 Exercise 78

Simplify the expression.

3m( 4+7 )−m 3 m \left(4 + 7\right) - m

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1.6 Exercise 80

Identify the number as rational, irrational, whole, or natural. Choose the most descriptive answer.

0

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1.6 Exercise 82

Identify the number as rational, irrational, whole, or natural. Choose the most descriptive answer.

11 \sqrt{11}

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1.6 Exercise 84

Simplify the expression.

4 5 4 3 \frac{4^{5}}{4^{3}}

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1.6 Exercise 86

Simplify the expression.

6 a 2 ⋅ a 0 2 a −4 \frac{6 a^{2} \cdot a^{0}}{2 a^{−4}}

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1.6 Exercise 88

Simplify the expression.

4 −2 x 3 y −3 2 x 0 \frac{4^{−2} x^{3} y^{−3}}{2 x^{0}}

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1.6 Exercise 90

Simplify the expression.

( 16 a 3 b 2 ) ( 4a b −1 ) −2 \left(\frac{16 a^{3}}{b^{2}}\right) \left(4 a b^{−1}\right)^{−2}

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1.6 Exercise 92

Simplify the expression.

Write the number in scientific notation: 16,340,000

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1.6 Exercise 94

Find the principal square root.

196 \sqrt{196}

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1.6 Exercise 96

Find the principal square root.

75 \sqrt{75}

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1.6 Exercise 98

Find the principal square root.

32 25 \sqrt{\frac{32}{25}}

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1.6 Exercise 100

Find the principal square root.

49 1250 \sqrt{\frac{49}{1250}}

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1.6 Exercise 102

Find the principal square root.

4 3 +6 3 4 \sqrt{3} + 6 \sqrt{3}

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1.6 Exercise 104

Find the principal square root.

−243 5 \sqrt[5]{−243}

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1.6 Exercise 106

Perform the given operations and simplify.

(3 x 3 +2x−1)+(4 x 2 −2x+7) \left(3 x^{3} + 2 x - 1\right) + \left(4 x^{2} - 2 x + 7\right)

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1.6 Exercise 108

Perform the given operations and simplify.

(2 x 2 +3x−6)+(3 x 2 −4x+9) \left(2 x^{2} + 3 x - 6\right) + \left(3 x^{2} - 4 x + 9\right)

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1.6 Exercise 110

Perform the given operations and simplify.

(k+3)(k−6) \left(k + 3\right) \left(k - 6\right)

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1.6 Exercise 112

Perform the given operations and simplify.

( x+1 )( x 2 +1 ) \left(x + 1\right) \left(x^{2} + 1\right)

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1.6 Exercise 114

Perform the given operations and simplify.

( a+2b )( 3a−b ) \left(a + 2 b\right) \left(3 a - b\right)

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1.6 Exercise 116

Find the greatest common factor.

81p+9pq−27 p 2 q 2 81 p + 9 p q - 27 p^{2} q^{2}

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1.6 Exercise 118

Find the greatest common factor.

88 a 3 b+4 a 2 b−144 a 2 88 a^{3} b + 4 a^{2} b - 144 a^{2}

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1.6 Exercise 120

Factor the polynomial.

8 a 2 +30a−27 8 a^{2} + 30 a - 27

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1.6 Exercise 122

Factor the polynomial.

x 2 +10x+25 x^{2} + 10 x + 25

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1.6 Exercise 124

Factor the polynomial.

4 h 2 −12hk+9 k 2 4 h^{2} - 12 h k + 9 k^{2}

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1.6 Exercise 126

Factor the polynomial.

p 3 +216 p^{3} + 216

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1.6 Exercise 128

Factor the polynomial.

64 q 3 −27 p 3 64 q^{3} - 27 p^{3}

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1.6 Exercise 130

Factor the polynomial.

3p ( p+3 ) 1 3 −8 ( p+3 ) 4 3 3 p \left(p + 3\right)^{\frac{1}{3}} −8 \left(p + 3\right)^{\frac{4}{3}}

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1.6 Exercise 132

Simplify the expression.

x 2 −x−12 x 2 −8x+16 \frac{x^{2} - x - 12}{x^{2} - 8 x + 16}

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1.6 Exercise 134

Simplify the expression.

2 a 2 −a−3 2 a 2 −6a−8 ⋅ 5 a 2 −19a−4 10 a 2 −13a−3 \frac{2 a^{2} - a - 3}{2 a^{2} - 6 a - 8} \cdot \frac{5 a^{2} - 19 a - 4}{10 a^{2} - 13 a - 3}

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1.6 Exercise 136

Simplify the expression.

m 2 +5m+6 2 m 2 −5m−3 ÷ 2 m 2 +3m−9 4 m 2 −4m−3 \frac{m^{2} + 5 m + 6}{2 m^{2} - 5 m - 3} \div \frac{2 m^{2} + 3 m - 9}{4 m^{2} - 4 m - 3}

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1.6 Exercise 138

Simplify the expression.

10 x + 6 y \frac{10}{x} + \frac{6}{y}

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1.6 Exercise 140

Simplify the expression.

1 d + 2 c 6c+12d dc \frac{\frac{1}{d} + \frac{2}{c}}{\frac{6 c + 12 d}{d c}}

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1.6 Exercise 142

Identify the number as rational, irrational, whole, or natural. Choose the most descriptive answer.

−13 −13

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1.6 Exercise 144

Evaluate the expression.

2(x+3)−12;x=2 2 \left(x + 3\right) - 12 ; x = 2

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1.6 Exercise 146

Evaluate the expression.

Write the number in standard notation: 3.1415× 10 6 3.1415 \times 10^{6}

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1.6 Exercise 148

Simplify the expression.

−2⋅ ( 2+3⋅2 ) 2 +144 −2 \cdot \left(2 + 3 \cdot 2\right)^{2} + 144

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1.6 Exercise 150

Simplify the expression.

3 5 ⋅ 3 −3 3^{5} \cdot 3^{−3}

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1.6 Exercise 152

Simplify the expression.

8 x 3 ( 2x ) 2 \frac{8 x^{3}}{\left(2 x\right)^{2}}

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1.6 Exercise 154

Simplify the expression.

441 \sqrt{441}

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1.6 Exercise 156

Simplify the expression.

9x 16 \sqrt{\frac{9 x}{16}}

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1.6 Exercise 158

Simplify the expression.

6 24 +7 54 −12 6 6 \sqrt{24} + 7 \sqrt{54} - 12 \sqrt{6}

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1.6 Exercise 160

Simplify the expression.

(13 q 3 +2 q 2 −3)−(6 q 2 +5q−3) \left(13 q^{3} + 2 q^{2} - 3\right) - \left(6 q^{2} + 5 q - 3\right)

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1.6 Exercise 162

Simplify the expression.

(n−2)( n 2 −4n+4) \left(n - 2\right) \left(n^{2} - 4 n + 4\right)

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1.6 Exercise 164

Factor the polynomial.

16 x 2 −81 16 x^{2} - 81

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1.6 Exercise 166

Factor the polynomial.

27 c 3 −1331 27 c^{3} - 1331

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1.6 Exercise 168

Simplify the expression.

2 z 2 +7z+3 z 2 −9 ⋅ 4 z 2 −15z+9 4 z 2 −1 \frac{2 z^{2} + 7 z + 3}{z^{2} - 9} \cdot \frac{4 z^{2} - 15 z + 9}{4 z^{2} - 1}

2.5

1.6 Exercise 170

Simplify the expression.

a 2b − 2b 9a 3a−2b 6a \frac{\frac{a}{2 b} - \frac{2 b}{9 a}}{\frac{3 a - 2 b}{6 a}}

2.5

Adapted from College Algebra © 2021 Rice University. Licensed under CC BY 4.0. Sources: College Algebra 2e, Algebra and Trigonometry 2e, Precalculus 2e, College Algebra with Corequisite Support 2e. Imported at revision 4922e46.