Algebraic Expressions

1.5. ALGEBRAIC EXPRESSIONS
1.5
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Algebraic Expressions
1. Use the associative property to regroup, then simplify.
−3(6a) = ((−3) · 6)a
= −18a
Apply the associative property.
Simplify.
3. Use the associative property to regroup, then simplify.
−9(6ab) = ((−9) · 6)ab
= −54ab
Apply the associative property.
Simplify.
5. Use the associative property to regroup, then simplify.
−7(3x2 ) = ((−7) · 3)x2
2
= −21x
Apply the associative property.
Simplify.
7. Use the distributive property to expand the expression, and then use order
of operations to simplify.
4(3x − 7y) = 4(3x) − 4(7y)
= 12x − 28y
Apply the distributive property.
Simplify.
9. Use the distributive property to expand the expression, and then use order
of operations to simplify.
−6(−y + 9) = −6(−y) + (−6)(9)
= 6y − 54
Apply the distributive property.
Simplify.
11. Use the distributive property to expand the expression, and then use order
of operations to simplify.
−9(s + 9) = −9(s) + (−9)(9)
= −9s − 81
Apply the distributive property.
Simplify.
13. To negate a sum, simply negate each term of the sum:
−(−3u − 6v + 8) = 3u + 6v − 8
Second Edition: 2012-2013
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CHAPTER 1. THE ARITHMETIC OF NUMBERS
15. Use the distributive property to expand the expression, and then use order
of operations to simplify.
−8(4u2 − 6v 2 ) = −8(4u2 ) − (−8)(6v 2 )
2
= −32u + 48v
2
Apply the distributive property.
Simplify.
17. To negate a sum, simply negate each term of the sum:
−(7u + 10v + 8) = −7u − 10v − 8
19. First use the distributive property to factor out the common variable part.
Then simplify.
−19x + 17x − 17x = (−19 + 17 − 17)x
= −19x
Distributive property.
Simplify.
21. First use the distributive property to factor out the common variable part.
Then simplify.
14x3 − 10x3 = (14 − 10)x3
3
= 4x
Distributive property.
Simplify.
23. First use the distributive property to factor out the common variable part.
Then simplify.
9y 2 x + 13y 2 x − 3y 2 x = (9 + 13 − 3)y 2 x
2
= 19y x
Distributive property.
Simplify.
25. First use the distributive property to factor out the common variable part.
Then simplify.
15m + 14m = (15 + 14)m
= 29m
Distributive property.
Simplify.
27. Combine like terms and write down the answer.
9 − 17m − m + 7 = 16 − 18m
Hint: 9 + 7 = 16 and −17m − m = −18m.
Second Edition: 2012-2013
1.5. ALGEBRAIC EXPRESSIONS
49
29. Combine like terms and write down the answer.
−6y 2 − 3x3 + 4y 2 + 3x3 = −2y 2
Hint: −6y 2 + 4y 2 = −2y 2 and −3x3 + 3x3 = 0.
31. Combine like terms and write down the answer.
−5m − 16 + 5 − 20m = −25m − 11
Hint: −5m − 20m = −25m and −16 + 5 = −11.
33. Combine like terms and write down the answer.
−16x2 y + 7y 3 − 12y 3 − 12x2 y = −28x2 y − 5y 3
Hint: −16x2 y − 12x2 y = −28x2 y and 7y 3 − 12y 3 = −5y 3 .
35. Combine like terms and write down the answer.
−14r + 16 − 7r − 17 = −21r − 1
Hint: −14r − 7r = −21r and 16 − 17 = −1.
37. Combine like terms and write down the answer.
14 − 16y − 10 − 13y = 4 − 29y
Hint: 14 − 10 = 4 and −16y − 13y = −29y.
39. Use the distributive property to expand the expression. Then combine like
terms mentally.
3 − (−5y + 1) = 3 + 5y − 1
= 2 + 5y
Distribute (negate the sum).
Combine like terms.
41. Use the distributive property to expand the expression. Then combine like
terms mentally.
− (9y 2 + 2x2 ) − 8(5y 2 − 6x2 )
= −9y 2 − 2x2 − 40y 2 + 48x2
2
2
= −49y + 46x
Distribute.
Combine like terms.
Second Edition: 2012-2013
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43. Use the distributive property to expand the expression. Then combine like
terms mentally.
2(10 − 6p) + 10(−2p + 5) = 20 − 12p − 20p + 50
= 70 − 32p
Distribute.
Combine like terms.
45. Use the distributive property to expand the expression. Then combine like
terms mentally.
4(−10n + 5) − 7(7n − 9) = −40n + 20 − 49n + 63
= −89n + 83
Distribute.
Combine like terms.
47. Use the distributive property to expand the expression. Then combine like
terms mentally.
−4x − 4 − (10x − 5) = −4x − 4 − 10x + 5
= −14x + 1
Distribute (negate the sum).
Combine like terms.
49. First use the distributive property to expand the expression. Then rearrange the terms and combine like terms.
−7 − (5 + 3x) = −7 − 5 − 3x
= −12 − 3x
Distribute (negate the sum).
Combine like terms.
51. Use the distributive property to expand the expression. Then combine like
terms mentally.
−8(−5y − 8) − 7(−2 + 9y) = 40y + 64 + 14 − 63y
= −23y + 78
Distribute.
Combine like terms.
53. Use the distributive property to expand the expression. Then combine like
terms mentally.
4(−7y 2 − 9x2 y) − 6(−5x2 y − 5y 2 )
= −28y 2 − 36x2 y + 30x2 y + 30y 2
2
2
= 2y − 6x y
Distribute.
Combine like terms.
55. Use the distributive property to expand the expression. Then combine like
terms mentally.
6s − 7 − (2 − 4s) = 6s − 7 − 2 + 4s
= 10s − 9
Second Edition: 2012-2013
Distribute (negate the sum).
Combine like terms.
1.5. ALGEBRAIC EXPRESSIONS
51
57. Use the distributive property to expand the expression. Then combine like
terms mentally.
9(9 − 10r) + (−8 − 2r) = 81 − 90r − 8 − 2r
= 73 − 92r
Distribute.
Combine like terms.
59. We analyze the expression inside the parentheses first. Multiply first,
distributing the −5.
−7x + 7[2x − 5[8x + 5]] = −7x + 7(2x − 40x − 25)
= −7x + 7(−38x − 25)
Next, distribute 7 and simplify.
= −7x − 266x − 175
= −273x − 175
61. We analyze the expression inside the parentheses first. Multiply first,
distributing the 2.
6x − 4[−3x + 2[5x − 7]] = 6x − 4(−3x + 10x − 14)
= 6x − 4(7x − 14)
Next, distribute −4 and simplify.
= 6x − 28x + 56
= −22x + 56
63. We analyze the expression inside the parentheses first. Multiply first,
distributing the −3.
−8x − 5[2x − 3[−4x + 9]] = −8x − 5(2x + 12x − 27)
= −8x − 5(14x − 27)
Next, distribute −5 and simplify.
= −8x − 70x + 135
= −78x + 135
Second Edition: 2012-2013