1. \(4x^2 – 5\)
2. \(4x – y\)
3. \(9x^4 – 16 y^6\)
4. \(h\)
5. \(x – \cfrac{1}{y^2}\)
6. \(a^2 + b^2 – c^2 + 2ab\)
7. \(16x^2 + 40x + 25\)
8. \(4x^2 -20xy +25y^2\)
9. \(x^2 – 2 + \cfrac{1}{x^2}\)
10. \(x^6 – 2 + \cfrac{1}{x^6}\)
11. \(64x^3+ 48x^2 y + 12xy^2 + y^3\)
12. \(a^6 + 3 a^4 b^2 + 3 a^2 b^4 + b^6\)
13. \(x^6 – 3x^4 y + 3x^2y^2 – y^3\)
14. \(x + 3 \sqrt[3]{x^2 y}+ 3 \sqrt[3]{x y^2} + y\)
15. \(x^4 – 50x^2 + 625\)
16. \(16x^4 – y^4\)
17. \(7 x^2 (x – 9)\)
18. \(4xy^2 z^2 \left( 2xz – 6y – x^2 y^2 z \right)\)
19. \((x – 2)^2 (x + 2)\)
20. \(4(y + 4)(y + 3x)\)
21. \((x – 1) \left( x y^2 + y^2 – 4 \right)\)
22. \((2x – 5y) \left( a^2 – 3b \right)\)
23. \((x + 8)(x – 4)\)
24. \((x – 5)(x + 1)\)
25. \((xy + 29)(y – 1)\)
26. \((x + 24)(x – 9)\)
27. \(\left( x^2 – 10 \right) \left( x^2 + 8 \right)\)
28. \((ab + 4)(ab – 3)\)
29. \((3x + 4)(x + 1)\)
30. \(5(y + 5)(y – 3)\)
31. \((ax + 2)(5ax – 6)\)
32. \((3x – 10)(3x + 5)\)
33. \(y^2 (x + 2)(4x + 3)\)
34. \(\left( 5x^2 – 1 \right)^2\)
35. \(\left( 5x + 6y^2 \right) \left( 5x – 6y^2 \right)\)
36. \(7x^2(3x + 1)(3x – 1)\)
37. \(5x^2 (3y + x)(3y – x)\)
38. \(\left( \cfrac{x}{6} + \cfrac{y}{5} \right) \left( \cfrac{x}{6} – \cfrac{y}{5} \right)\)
39. \(\left( 4x^n + \cfrac{1}{7} \right) \left( 4x^n – \cfrac{1}{7} \right)\)
40. \((a – b + 3)(a – b – 3)\)
41. \(4ab\)
42. \((x + y – 3)(x – y + 1)\)
43. \((x + y + 3)(x – y – 3)\)
44. \((5a – b)(a – 5b)\)
45. \(\left( a^2 – 1 \right)^2 \)
46. \((4x – 3y)^2\)
47. \( \left( 20x^2 + 1 \right)^2\)
48. \(\left( \cfrac{x}{3} + 1 \right)^2\)
49. \( \left( \cfrac{2x}{5} – \cfrac{1}{4}\right)^2\)
50. \((2x – y) \left( 4x^2 + 2xy + y^2 \right)\)
51. \((3a + 4b) \left( 9a^2 – 12ab + 16b^2 \right)\)
52. \(5(xy + 1) \left( x^2y^2 – xy + 1 \right)\)
53. \(x^2 (x – 5) \left( x^2 + 5x + 25 \right)\)
54. \((x + y – 1)\left( x^2 + 2xy + y^2 + x + y + 1 \right)\)
55. \((x – y – 2) \left(x^2 – 2xy + y^2 + 2x – 2y + 4 \right)\)
56. \(9 \left( x^2 – x + 1 \right)\)
57. \(4ab – 3\)
58. \(-x\)
59. \(a – 1\)
60. \(\cfrac{x – 5}{x – 2}\)
61. \(2x + 4 \)
62. \(\cfrac{x – 1}{2x + 2}\)
63. \(\cfrac{x-y}{x+y}\)
64. \(\cfrac{x – 2y}{ x^2 + 2xy + 4y^2}\)
65. \(\cfrac{3 – a}{ 9 + 3a + a^2}\)
66. \(\cfrac{1}{x-1}\)
67. \(\cfrac{4y +1}{ y^2 + 6y}\)
68. \(\cfrac{x + y}{6 + x}\)
69. \(- 2 – 2 \sqrt{2}\)
70. \(\sqrt{3 + h} + \sqrt{3}\)
71. \(a \sqrt{a + 1} + a \sqrt{a – 1}\)
72. \(- \cfrac{21 + 9\sqrt{6}}{5}\)
73. \(\cfrac{ x – \sqrt{a x} – 2a }{x – 4a}\)
74. \(\cfrac{ \sqrt{x – 3} + \sqrt{x – 13}}{2} \)
75. \(\cfrac{ \sqrt[3]{49}- \sqrt[3]{14} + \sqrt[3]{4} }{3}\)
76. \(8 \sqrt[3]{x^2} + 4 \sqrt[3]{x} + 2\)
77. \(8 \sqrt[3]{(x – 1)^2} – 12 \sqrt[3]{ x^2 – x } + 18 \sqrt[3]{x^2}\)
78. \(3 \sqrt[3]{x} – 3 \sqrt[3]{3y}\)
79. \(\left( \sqrt{ 2 – \sqrt[3]{x}} \right) \left( 4 + 2 \sqrt[3]{x} + \sqrt[3]{x^2} \right)\)
80. \(\sqrt{ 2 \sqrt[3]{x^2} + \sqrt{2} x } – \sqrt{ 4 \sqrt{x} + \sqrt{8}}\)
81. \(\cfrac{1}{3 – \sqrt{5}}\)
82. \(\cfrac{1}{\sqrt{a + 2} + \sqrt{a}} \)
83. \(\cfrac{ 1 }{ \sqrt{ a – 1 + h } + \sqrt{a – 1}}\)
84. \(\cfrac{5 a^2 – a}{a^2 – 1} \)
85. \(\cfrac{ 4xy }{ x^2 – y^2 }\)
86. \(\cfrac{x + 1}{x + 3}\)
87. \(\cfrac{x – 3}{x^2 – 1}\)
88. \(\cfrac{3x}{x^2 – 1}\)
89. \(\cfrac{3x^2 + 3x – 24}{ x^3 – 3x^2 – 9x – 5 }\)
90. \(\cfrac{ x + 2 }{ x^2 + 8x + 7 } \)
91. \(\cfrac{x^2 y – x^3} { y^3 – x^2 y } \)
92. \(\cfrac{ 3 x^2 + 2x }{x – 4}\)
93. \(\cfrac{x – 2}{a – 1} \)
94. \(\cfrac{ x^3 + 8y^3 – 3y x^2 – 6xy^2 }{ x^2 – 2xy – 8 y^2}\)
95. \(\cfrac{ a^3 – a^2 b – 6 a b^2 } { a^2 b – 4 b^3 } \)
96. \(\cfrac{x^3 + x^2 + x}{ x + 2}\)
97. \(\cfrac{5x^2 + x} {2x + 3}\)
98. \(\cfrac{x – 1}{x}\)
99. \(\cfrac{ 3x – 9 }{x^2 + 2x}\)
100. \(\cfrac{ b^2 + ab + a^2}{b}\)
101. \(x^2 + 6x\)
102. \(\cfrac{y}{x}\)
103. \(\cfrac{x – 1}{ x^2 + 1}\)
104. \(\cfrac{ a }{ a^2 + 2}\)
105. \(x^2 + x + 1\)
106. \(\cfrac{a^2 + ab + b^2}{a + b}\)
107. \(\cfrac{a^3}{ a^2 + b^2 }\)
108. \(x^2\)
109. \(\cfrac{a+1}{a^2 + 5a – 14} \)

For exercises 1 through 16, use the Special Product Formulas to find each product.

1. \[ \left( 2x+\sqrt{5} \right)\left( 2x-\sqrt{5} \right) \]
2. \[ \left( 2\sqrt{x}+\sqrt{y} \right)\left( 2\sqrt{x}-\sqrt{y} \right) \]
3. \[ \left( 3x^2+4y^3\right)\left( 3x^2-4y^3 \right) \]
4. \[ \left( \sqrt{h+1}+1 \right) \left( \sqrt{h+1}-1 \right) \]
5. \[ \left( \sqrt{x}+ \frac{1}{y} \right) \left( \sqrt{x} – \frac{1}{y} \right) \]
6. \[ (a+b+c)(a+b-c) \]
7. \[ (4x+5)^2 \]
8. \[ (2x-5y)^2 \]
9. \[ \left( x-x^{-1} \right)^2 \]
10. \[ \left( x^3-x^{-3} \right)^2 \]
11. \[ (4x+y)^3 \]
12. \[ \left( a^2 + b^2\right)^3 \]
13. \[ \left( x^2-y \right)^3 \]
14. \[ \left( \sqrt[3]{x}+\sqrt[3]{y} \right)^3 \]
15. \[ (x-5)^2(x+5)^2 \]
16. \[ (2x-y)(2x+y)\left( 4x^2+y^2 \right) \]

For exercises 17 through 56, factorize the expression.

17. \[ 7x^3-63x^2 \]
18. \[ 8x^2y^2z^3-24xy^3z^2-4x^3y^4z^3 \]
19. \[ x^3-2x^2-4x+8 \]
20. \[ 4y^2+16y+12xy+48x \]
21. \[ x^2y^2-y^2-4x+4 \]
22. \[ 2a^2x-5a^2y+15by-6bx \]
23. \[ x^2+2x-48 \]
24. \[ x^2-4x-5 \]
25. \[ y^2+28y-29 \]
26. \[ x^2+15x-216 \]
27. \[ x^4-2x^2-80 \]
28. \[ a^2b^2+ab-12 \]
29. \[ 3x^2 + 7x + 4 \]
30. \[ 5y^2 + 10y – 75 \]
31. \[ 5a^2x^2 + 4ax – 12 \]
32. \[ 9x^2 – 15x – 50 \]
33. \[ 4x^2y^2 + 11xy^2 + 6y^2 \]
34. \[ 25x^4 – 10x^2 + 1 \]
35. \[ 25x^2 – 36y^4 \]
36. \[ 63x^4 – 7x^2 \]
37. \[ 45x^2y^2 – 5x^4 \]
38. \[ \frac{x^2}{36} – \frac{y^2}{25} \]
39. \[ 16x^{2n} – \frac{1}{49} \]
40. \[ (a – b)^2 – 9 \]
41. \[ (a + b)^2 – (a – b)^2 \]
42. \[ (x – 1)^2 – (y – 2)^2 \]
43. \[ x^2 – y^2 – 6y – 9 \]
44. \[ 9(a – b)^2 – 4(a + b)^2 \]
45. \[ a^4 – 2a^2 + 1 \]
46. \[ 16x^2 – 24xy + 9y^2 \]
47. \[ 400x^4+ 40x^2 + 1 \]
48. \[ \frac{x^2}{9} + \frac{2x}{3} + 1 \]
49. \[ \frac{4x^2}{25} – \frac{x}{5} + \frac{1}{16} \]
50. \[ 8x^3- y^3 \]
51. \[ 27a^3 + 64b^3 \]
52. \[ 5x^3y^3 + 5 \]
53. \[ x^5- 125x^2 \]
54. \[ (x + y)^3 – 1 \]
55. \[ (x – y)^3 – 8 \]
56. \[ (x + 1)^3 – (x – 2)^3 \]

For exercises 57 through 68, simplify the given fraction.

57. \[ \frac{60a^3b^2-45a^2b}{15a^2b} \]
58. \[ \frac{x^2-3x}{3-x} \]
59. \[ \frac{a^2-1}{a+1} \]
60. \[ \frac{x^2-x-20}{x^2+2x-8} \]
61. \[ \frac{2x^2+x-6}{2x-3} \]
62. \[ \frac{x^2+x-2}{2x^2+6x+4} \]
63. \[ \frac{x^2-y^2}{x^2+2xy+y^2} \]
64. \[ \frac{x^2-4xy+4y^2}{x^3-8y^3} \]
65. \[ \frac{(3-a)^2}{27-a^3} \]
66. \[ \frac{x^3+1}{x^4-x^3+x-1} \]
67. \[ \frac{y+8y^2+16y^3}{6y^2+25y^3+4y^4} \]
68. \[ \frac{x^2-y^2}{x^2-6y-xy+6x} \]

For exercises 69 through 80, rationalize the denominator.

69. \[ \frac{2}{1-\sqrt{2}} \]
70. \[ \frac{h}{\sqrt{3+h}-\sqrt{3}} \]
71. \[ \frac{2a}{\sqrt{a+1}-\sqrt{a-1}} \]
72. \[ \frac{3\sqrt{2}}{7\sqrt{2}-6\sqrt{3}} \]
73. \[ \frac{\sqrt{x}+\sqrt{a}}{\sqrt{x}+2\sqrt{a}} \]
74. \[ \frac{5}{\sqrt{x-3}-\sqrt{x-13}} \]
75. \[ \frac{3}{\sqrt[3]{7}+\sqrt[3]{2}} \]
76. \[ \frac{16x-2}{2\sqrt[3]{x}-1} \]
77. \[ \frac{70x-16}{2\sqrt[3]{x-1}+3\sqrt[3]{x}} \]
78. \[ \frac{3x-9y}{\sqrt[3]{x^2}+\sqrt[3]{3xy}+\sqrt[3]{9y^2}} \]
79. \[ \frac{8-x}{\sqrt{2-\sqrt[3]{x}}} \]
80. \[ \frac{2x-1}{\sqrt{2\sqrt{x}+\sqrt{2}}} \]

For exercises 81 through 83, rationalize the numerator.

81. \[ \frac{3+\sqrt{5}}{4} \]
82. \[ \frac{\sqrt{a+2}-\sqrt{a}}{2} \]
83. \[ \frac{\sqrt{a-1+h}-\sqrt{a-1}}{h} \]

For exercises 84 through 104, perform the given operations and simplify.

84. \[ \frac{3a}{a+1} + \frac{2a}{a-1} \]
85. \[ \frac{x+y}{x-y} – \frac{x-y}{x+y} \]
86. \[ \frac{12}{x^2-9} – \frac{2}{x-3} + 1 \]
87. \[ \frac{x-2}{x^2-x-2} – \frac{2}{x^2-1} \]
88. \[ \frac{1}{x+1} + \frac{2}{x-1} – \frac{1}{x^2-1} \]
89. \[ \frac{x+5}{x^2+2x+1} + \frac{x}{x^2-4x-5} + \frac{1}{x-5} \]
    \[ \begin{aligned} \frac{x+5}{x^2+2x+1} &+ \frac{x}{x^2-4x-5} \\[1em] &+ \frac{1}{x-5} \end{aligned} \]
90. \[ \frac{x}{x^2-x-2} – \frac{6}{x^2+5x-14} – \frac{1}{x^2+8x+7} \]
    \[ \begin{aligned} \frac{x}{x^2-x-2} &- \frac{6}{x^2+5x-14} \\[1em] &- \frac{1}{x^2+8x+7} \end{aligned} \]
91. \[ \frac{x^2}{y^2-x^2} \times \frac{xy-x^2}{xy} \]
92. \[ \frac{x^2+4x}{3x-2} \times \frac{9x^2-4}{x^2-16} \]
93. \[ \frac{x^3-8}{a^3-1} \times \frac{a^2+a+1}{x^2+2x+4} \]
94. \[ \begin{aligned} \frac{ x^2 + xy – 2y^2 }{ x^2 – 2xy – 8 y^2 } &\times \frac{ x^2 + 2 xy }{ x^2 + 4 xy} \\[1em] &\times \frac{ x^2 – 16 y^2 }{ x + 2y} \end{aligned} \]
    \[ \frac{ x^2 + xy – 2y^2 }{ x^2 – 2xy – 8 y^2 } \times \frac{ x^2 + 2 xy }{ x^2 + 4 xy} \times \frac{ x^2 – 16 y^2 }{ x + 2y} \]
95. \[ \frac{a^2-ab-6b^2}{b^2+ab} \div \frac{a^2-4b^2}{a^2+ab} \]
96. \[ \frac{x^4-x}{x^2+6x+8} \div \frac{2x^2-x-1}{2x^2+9x+4} \]
97. \[ \frac{25x^3-x}{25x^2-10x+1} \div \frac{6x^2+13x+6}{15x^2+7x-2} \]
    \[ \begin{aligned} &\frac{25x^3-x}{25x^2-10x+1} \\[1em] &\hspace{3em} \div \frac{6x^2+13x+6}{15x^2+7x-2} \end{aligned} \]
98. \[ \left( \frac{x+1}{3x-3} \times \frac{6x-6}{2x+4} \right) \div \frac{x^2+x}{x^2+x-2} \]
    \[ \begin{aligned} &\left( \frac{x+1}{3x-3} \times \frac{6x-6}{2x+4} \right) \\[1em] &\hspace{3em} \div \frac{x^2+x}{x^2+x-2} \end{aligned} \]
99. \[ \frac{3x^2+3}{2x-4} \div \left( \frac{3x+6}{2x-6} \times \frac{x^3+x}{3x-6} \right) \]
    \[ \begin{aligned} &\frac{3x^2+3}{2x-4} \\[1em] &\hspace{2em} \div \left( \frac{3x+6}{2x-6} \times \frac{x^3+x}{3x-6} \right) \end{aligned} \]
100. \[ \left( 1-\frac{a^3}{b^3} \right) \left( b+\frac{ab}{b-a} \right) \]
101. \[ \left( x+\frac{4x^2+20x}{x^2-25} \right) \left( x+2-\frac{28}{x-1} \right) \]
     \[ \begin{aligned} &\left( x+\frac{4x^2+20x}{x^2-25} \right) \\[1em] &\hspace{2em} \times \left( x+2-\frac{28}{x-1} \right) \end{aligned} \]
102. \[ \left( \frac{x^2}{x^2-y^2}-1 \right) \left( \frac{x}{y}-1 \right) \left( \frac{y}{x}+1 \right) \]
     \[ \begin{aligned} &\left( \frac{x^2}{x^2-y^2}-1 \right) \left( \frac{x}{y}-1 \right) \\[1em] &\hspace{2em} \times \left( \frac{y}{x}+1 \right) \end{aligned} \]
103. \[ \left( \frac{x^2}{x+1} -x+1 \right) \div \left( \frac{2}{x^2-1} +1 \right) \]
     \[ \begin{aligned} &\left( \frac{x^2}{x+1} -x+1 \right) \\[1em] &\hspace{2em}\div \left( \frac{2}{x^2-1} +1 \right) \end{aligned} \]
104. \[ \left( \frac{2a+1}{a^2+2} -a \right) \div \left( \frac{a+1}{a} -a^2-1 \right) \]
     \[ \begin{aligned} &\left( \frac{2a+1}{a^2+2} -a \right) \\[1em] &\hspace{2em} \div \left( \frac{a+1}{a} -a^2-1 \right) \end{aligned} \]

Simplify the given compound fractions.

105. \[ \cfrac{ \cfrac{1}{x} -x^2 }{ \cfrac{1}{x} -1 } \]
106. \[ \cfrac{ \cfrac{a}{b^2} – \cfrac{b}{a^2} } { \cfrac{1}{b^2} – \cfrac{1}{a^2} } \]
107. \[ a- \cfrac{b} { \cfrac{a}{b} + \cfrac{b}{a} } \]
108. \[ 1- \cfrac{1}{ 1- \cfrac{1} { 1- \cfrac{1}{x^2} } } \]
109. \[ \cfrac{ 1- \cfrac{1}{a-2} }{ a+3 – \cfrac{24}{a+1} } \]