1. Simplify the following expression.
%7d%7b5x%7d)
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2. Factor the following expression completely.
49x10 - x8
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A. |
(7x5
- x2)2 |
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B. |
x8(49x2 -
1) |
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C. |
x8(7x - 1)(7x +
1) |
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D. |
x8(7x - 1)2 |
3. Simplify.
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4. Simplify the following expression to an answer with only positive exponents.
\left(-4x%5e%7b3%7dz%5e%7b-2%7d\right)%7d%7by%5e%7b8%7dz%5e%7b3%7d%7d)
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5. Simplify.
%7d%7bx%5e%7b2%7dy%5e%7b4%7d%7d\%20-\%20\frac%7b4x%5e%7b2%7d\%20-\%202x%5e%7b2%7d%7d%7bx%5e%7b3%7dy%7d)
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6. Factor the following expression completely.
x4 - 16
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A. |
(x
- 2)(x + 2)(x2 + 4) |
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B. |
(x
- 2)(x3 + 8) |
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C. |
(x2
- 4)(x2 + 4) |
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D. |
(x
- 2)(x + 2)(x - 2)(x + 2) |
7. Simplify the following expression.
(x\%20+\%2015))
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8. Factor the following expression completely.
36x8 - x6
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A. |
x6(6x - 1)2 |
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B. |
x6(36x2 -
1) |
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C. |
x6(6x - 1)(6x +
1) |
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D. |
(6x4
- x2)2 |
9. Simplify the following expression.
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10. Simplify.
%7d%7b4%7d)
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11. Factor the following expression completely.
16x9 - x5
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A. |
(4x7
- x3)2 |
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B. |
x5(2x - 1)2(4x2
+ 1) |
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C. |
x5(2x - 1)(2x +
1)(4x2 + 1) |
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D. |
x5(4x2 -
1)(4x2 + 1) |
12. Simplify the following expression.
)
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13. Factor the following expression completely.
16x2 - 81y2
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A. |
(16x
- 81y)2 |
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B. |
(4x
+ 9y)(4x - 9y) |
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C. |
(4x
- 9y)2 |
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D. |
(16x
+ 81y)(16x - 81y) |
14. Factor the following expression completely.
81x7 - x3
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A. |
x3(9x2 -
1)(9x2 + 1) |
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B. |
x(9x4 - 1)2 |
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C. |
x3(81x4 -
1) |
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D. |
x3(3x - 1)(3x +
1)(9x2 + 1) |
15. Simplify the following expression to an answer with only positive exponents.
\left(7x%5e%7b-2%7dy%5e%7b2%7d\right))
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16. Factor the following expression completely.
x8 - y8
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A. |
(x
- y)(x + y)(x2 + y2)(x4
+ y4) |
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B. |
(x
- y)8 |
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C. |
(x4
+ y4)(x4 - y4) |
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D. |
(x2
+ y2)2 (x2 - y2)2 |
17. Factor the following expression completely.
25x12 - x10
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A. |
x10(5x - 1)(5x +
1) |
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B. |
(5x6
- x2)2 |
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C. |
x10(5x - 1)2 |
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D. |
x10(25x2 -
1) |
18. Factor the following expression completely.
64x8 - x6
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A. |
x6(64x2 -
1) |
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B. |
(8x4
- x2)2 |
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C. |
x6(8x - 1)(8x +
1) |
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D. |
x6(8x - 1)2 |
19. Simplify the following expression.
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20. Simplify the following expression.
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21. Factor the following expression completely.
9x2 - 64y2
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A. |
(3x
- 8y)2 |
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B. |
(9x
- 64y)2 |
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C. |
(9x
+ 64y)(9x - 64y) |
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D. |
(3x
+ 8y)(3x - 8y) |
22. Simplify the following expression to an answer with only positive exponents.
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23. Factor the following expression completely.
x4 - 1
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A. |
(x
- 1)(x + 1)(x2 + 1) |
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B. |
(x
- 1)(x3 + 1) |
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C. |
(x
- 1)(x + 1)(x - 1)(x + 1) |
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D. |
(x2
- 1)(x2 + 1) |
24. Simplify the following expression.
(x\%20+\%201)%7d)
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25. Factor the following expression completely.
4x2 - 49y2
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A. |
(2x
- 7y)2 |
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B. |
(2x
+ 7y)(2x - 7y) |
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C. |
(4x
+ 49y)(4x - 49y) |
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D. |
(4x
- 49y)2 |
26. Simplify the following expression to an answer with only positive exponents.
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27. Factor the following expression completely.
x4 - y4
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A. |
(x
- y)2 (x + y)2 |
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B. |
(x
- y)(x + y)(x2 + y2) |
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C. |
(x
- y)4 |
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D. |
(x2
- y2)(x2 + y2) |
28. Factor the following expression completely.
x4 - 81
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A. |
(x
- 3)(x + 3)(x - 3)(x + 3) |
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B. |
(x2
- 9)(x2 + 9) |
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C. |
(x
- 3)(x3 + 27) |
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D. |
(x
- 3)(x + 3)(x2 + 9) |
29. Simplify the following expression.
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30. Simplify the following expression:
%5e%7b2%7d)
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31. Simplify the following expression.
%5e%7b2%7d%7d%7by%5e%7b2%7d%7d%20\%20+%20\%204)
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32. Simplify the following expression:
%5e%7b2%7d)
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33. Simplify the following expression.
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34. Factor the following expression completely.
25x2 - 81y2
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A. |
(25x
- 81y)2 |
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B. |
(5x
+ 9y)(5x - 9y) |
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C. |
(5x
- 9y)2 |
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D. |
(25x
+ 81y)(25x - 81y) |
35. Factor the following expression completely.
256x10 - x6
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A. |
x6(4x - 1)(4x +
1)(16x2 + 1) |
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B. |
x6(4x - 1)2(16x2
+ 1) |
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C. |
x6(256x4 -
1) |
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D. |
x6(16x2 -
1)(16x2 + 1) |
36. Simplify.
\%20+\%208x(y\%20+\%205y)%7d%7bx%5e%7b2%7dy%7d)
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37. Simplify the following expression.
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38. Simplify the following expression.
%20\%20+%20\%2042z)
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39. Simplify the following expression.
%5e%7b2%7d%20\%20+%20\%204x)
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40. Simplify the following expression.
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Answers
1. D
2. C
3. A
4. B
5. C
6. A
7. D
8. C
9. D
10. D
11. C
12. B
13. B
14. D
15. C
16. A
17. A
18. C
19. C
20. C
21. D
22. B
23. A
24. C
25. B
26. D
27. B
28. D
29. C
30. D
31. C
32. A
33. A
34. B
35. A
36. D
37. B
38. D
39. D
40. C
Explanations
1.
%7d%7b5x%7d%20&=&%20\frac%7b2\left(3y%20\%20+%20\%2013z\right)%7d%7b5%7d\\\vspace*%7b9%20\hspace*%7b-10%7d%7d\\&=&%20\frac%7b6y%20\%20+%20\%2026z%7d%7b5%7d\end%7beqnarray%7d)
2. First, factor out x8, then factor using the difference of two squares.
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49x10
- x8 |
= |
x8(49x2 -
1) |
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= |
x8(7x - 1)(7x +
1) |
3. Simplify the first part of the expression.
Simplify the second part of the expression.
Combine like terms.
4.
\left(-4x%5e%7b3%7dz%5e%7b-2%7d\right)%7d%7by%5e%7b8%7dz%5e%7b3%7d%7d%20&=&%20\frac%7b\left(2%20\%20\cdot%20\%20-4\right)\left(x%5e%7b2%7d%20\%20\cdot%20\%20x%5e%7b3%7d\right)\left(y%5e%7b6%7d\right)\left(z%5e%7b-2%7d\right)%7d%7by%5e%7b8%7dz%5e%7b3)
5. Simplify the first part of the expression.
%7d%7bx%5e%7b2%7dy%5e%7b4%7d%7d\%20=\%20\frac%7b35xy%5e%7b3%7d%7d%7bx%5e%7b2%7dy%5e%7b4%7d%7d\%20=\%20\frac%7b35%7d%7bxy%7d)
Simplify the second part of the expression.
Combine like terms.
6. Factor using the difference of two squares.
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x4 - 16 |
= |
(x2
- 4)(x2 + 4) |
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= |
(x
- 2)(x + 2)(x2 + 4) |
7. First, factor the polynomial in the numerator.
(x\%20+\%2015)%7d\%20=\%20\frac%7b(5x\%20-\%2015)(x\%20-\%208)%7d%7b(x\%20-\%208)(x\%20+\%2015)%7d)
Then, eliminate the common factors.
\sout%7b(x\%20-\%208)%7d%7d%7b\sout%7b(x\%20-\%208)%7d(x\%20+\%2015)%7d\%20=\%20\frac%7b5x\%20-\%2015%7d%7bx\%20+\%2015%7d)
8. First, factor out x6, then factor using the difference of two squares.
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36x8
- x6 |
= |
x6(36x2 -
1) |
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= |
x6(6x - 1)(6x +
1) |
9. First, factor the polynomials in the numerator and the denominator.
(x\%20+\%2010)%7d%7b(2x\%20+\%201)(x\%20-\%2010)%7d)
Then, eliminate the common factors.
%7d(x\%20+\%2010)%7d%7b(2x\%20+\%201)\sout%7b(x\%20-\%2010)%7d%7d\%20=\%20\frac%7bx\%20+\%2010%7d%7b2x\%20+\%201%7d)
10. Reduce the fraction and combine like terms.
%7d%7b4%7d\%20&=&\%20\frac%7bx\%20+\%207x\%20+\%20y%7d%7b2%7d\\\vspace*%7b9\hspace*%7b-10%7d%7d\\&=&\%20\frac%7b8x\%20+\%20y%7d%7b2%7d\end%7beqnarray%7d)
11. First, factor out x5, then factor using the difference of two squares.
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16x9
- x5 |
= |
x5(16x4 -
1) |
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= |
x5(4x2 -
1)(4x2 + 1) |
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= |
x5(2x - 1)(2x +
1)(4x2 + 1) |
12.
%20&=&%20x%5e%7b8%7d\left(2x%5e%7b9%7d\right)%20\%20+%20\%20x%5e%7b8%7d\left(2x\right)\\\vspace*%7b9%20\hspace*%7b-10%7d%7d\\&=&%202x%5e%7b17%7d%20\%20+%20\%202x%5e%7b9%7d\end%7beqnarray%7d)
13. Factor using the difference of two squares.
|
16x2 - 81y2 |
= |
(4x)2
- (9y)2 |
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= |
(4x
+ 9y)(4x - 9y) |
14. First, factor out x3, then factor using the difference of two squares.
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81x7
- x3 |
= |
x3(81x4 -
1) |
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= |
x3(9x2 -
1)(9x2 + 1) |
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= |
x3(3x - 1)(3x +
1)(9x2 + 1) |
15.
\left(7x%5e%7b-2%7dy%5e%7b2%7d\right)%20&=&%20\left(-4%20\%20\cdot%20\%207\right)\left(x%5e%7b5%7d%20\%20\cdot%20\%20x%5e%7b-2%7d\right)\left(y%5e%7b3%7d%20\%20\cdot%20\%20y%5e%7b2%7d\right)\\\vspace*%7b9%20\hspace*%7b-10%7d%7d\\&=&%20\left)
16. Factor using the difference of two squares.
|
x8 - y8 |
= |
(x4
- y4)(x4 + y4) |
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= |
(x2
- y2)(x2 + y2)(x4
+ y4) |
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= |
(x
- y)(x + y)(x2 + y2)(x4
+ y4) |
17. First, factor out x10, then factor using the difference of two squares.
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25x12
- x10 |
= |
x10(25x2 -
1) |
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= |
x10(5x - 1)(5x +
1) |
18. First, factor out x6, then factor using the difference of two squares.
|
64x8
- x6 |
= |
x6(64x2 -
1) |
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= |
x6(8x - 1)(8x +
1) |
19. First, factor the polynomials in the numerator and the denominator.
(5x\%20-\%201)%7d%7b-x(5x\%20-\%201)%7d)
Then, eliminate the common factors.
\sout%7b(5x\%20-\%201)%7d%7d%7b-x\sout%7b(5x\%20-\%201)%7d%7d\%20=\%20-\,\frac%7bx\%20-\%2010%7d%7bx%7d)
20. First, factor the polynomials in the denominator.
(x\%20-\%205)%7d)
Then, eliminate the common factors.
\sout%7b(x\%20-\%205)%7d%7d\%20=\%20\frac%7b1%7d%7b-8x\%20-\%2015%7d)
21. Factor using the difference of two squares.
|
9x2 - 64y2 |
= |
(3x)2
- (8y)2 |
|
= |
(3x
+ 8y)(3x - 8y) |
22.
\left(m%5e%7b-7%7d%20\%20\cdot%20\%20m%5e%7b-3%7d\right)\left(n%5e%7b5%7d%20\%20\cdot%20\%20n%5e%7b-3%7d\right)\\\vspace*%7b9%20\hspace*%7b-10%7d%7d\\&=&%20\left(2\right)\left(m%5e%7b-)
23. Factor using the difference of two squares.
|
x4 - 1 |
= |
(x2
- 1)(x2 + 1) |
|
= |
(x
- 1)(x + 1)(x2 + 1) |
24. First, factor the polynomial in the numerator.
(x\%20+\%201)%7d\%20=\%20\frac%7b(x\%20+\%2010)(x\%20-\%205)%7d%7b(x\%20+\%2010)(x\%20+\%201)%7d)
Then, eliminate the common factors.
%7d(x\%20-\%205)%7d%7b\sout%7b(x\%20+\%2010)%7d(x\%20+\%201)%7d\%20=\%20\frac%7bx\%20-\%205%7d%7bx\%20+\%201%7d)
25. Factor using the difference of two squares.
|
4x2 - 49y2 |
= |
(2x)2
- (7y)2 |
|
= |
(2x
+ 7y)(2x - 7y) |
26.
27. Factor using the difference of two squares.
|
x4 - y4 |
= |
(x2
- y2)(x2 + y2) |
|
= |
(x
- y)(x + y)(x2 + y2) |
28. Factor using the difference of two squares.
|
x4 - 81 |
= |
(x2
- 9)(x2 + 9) |
|
= |
(x
- 3)(x + 3)(x2 + 9) |
29. First, factor the polynomials in the numerator and the denominator.
(x\%20-\%2013)%7d%7b(x\%20-\%208)(x\%20+\%2013)%7d)
Then, eliminate the common factors.
%7d(x\%20-\%2013)%7d%7b\sout%7b(x\%20-\%208)%7d(x\%20+\%2013)%7d\%20=\%20\frac%7bx\%20-\%2013%7d%7bx\%20+\%2013%7d)
30.
|
|
To simplify the expression, begin by rewriting the squared binomial without an exponent.
\left(\sqrt%7b13x%7d%20\%20+%20\%20\sqrt%7b6x%7d\right))
Multiply the binomials and simplify by combining like terms.
\left(\sqrt%7b13x%7d%20\%20+%20\%20\sqrt%7b6x%7d\right)%20&=&%2013x%20\%20+%20\%20\sqrt%7b78x%5e%7b2%7d%7d%20\%20+%20\%20\sqrt%7b78x%5e%7b2%7d%7d%20\%20+%20\%206x\\\vspace*%7b9%20\hspace*%7b-10%7d%7d\\&=&%2013x%20\%20+%20\%20x\sqrt%7b78)
31.
%5e%7b2%7d%7d%7by%5e%7b2%7d%7d%20\%20+%20\%204%20&=&%20\frac%7b\left(7x%20\%20+%20\%205y\right)\left(7x%20\%20+%20\%205y\right)%7d%7by%5e%7b2%7d%7d%20\%20+%20\%204\\\vspace*%7b9%20\hspace*%7b-10%7d%7d\\&=&%20\frac%7b49x%5e%7b2%7d%20\%20+%20\%2070xy%20\%20+%20\%20)
32. To simplify the expression, begin by rewriting the squared binomial without an exponent.
\left(x%20\%20-%20\%20\sqrt%7b7x%7d\right))
Multiply the binomials and simplify by combining like terms.
\left(x%20\%20-%20\%20\sqrt%7b7x%7d\right)%20&=&%20x%5e%7b2%7d%20\%20-%20\%20x\sqrt%7b7x%7d%20\%20-%20\%20x\sqrt%7b7x%7d%20\%20+%20\%207x\\\vspace*%7b9%20\hspace*%7b-10%7d%7d\\&=&%20x%5e%7b2%7d%20\%20-%20\%202x\sqrt%7b7x%7d%20\%20+%20\%207x\end%7beqnarr)
33. First, factor the polynomials in the numerator and the denominator.
%7d%7b(x\%20-\%2021)(x\%20+\%2020)%7d)
Then, eliminate the common factors.
%7d%7d%7b(x\%20-\%2021)\sout%7b(x\%20+\%2020)%7d%7d\%20=\%20\frac%7b-1%7d%7bx\%20-\%2021%7d)
34. Factor using the difference of two squares.
|
25x2 - 81y2 |
= |
(5x)2
- (9y)2 |
|
= |
(5x
+ 9y)(5x - 9y) |
35. First, factor out x6, then factor using the difference of two squares.
|
256x10
- x6 |
= |
x6(256x4 -
1) |
|
= |
x6(16x2 -
1)(16x2 + 1) |
|
|
= |
x6(4x - 1)(4x +
1)(16x2 + 1) |
36. Simplify inside the parentheses and distribute across the parentheses. Then, reduce the fraction.
\%20+\%208x(y\%20+\%205y)%7d%7bx%5e%7b2%7dy%7d\%20&=&\%20\frac%7b14x\%20+\%208x(6y)%7d%7bx%5e%7b2%7dy%7d\\\vspace*%7b9\hspace*%7b-10%7d%7d\\&=&\%20\frac%7b14x\%20+\%2048xy%7d%7bx%5e%7b2%7dy%7d\\\vspace*%7b9\hspace*%7b-10%7d%7d\\&=&\%20\frac%7b14\%20+\%2048y%7d%7bxy%7d\e)
37. First, factor the polynomial in the numerator.
(11x\%20-\%205)%7d%7b11x\%20-\%205%7d)
Then, eliminate the common factors.
\sout%7b(11x\%20-\%205)%7d%7d%7b\sout%7b11x\%20-\%205%7d%7d\%20=\%20-11x\%20-\%205\\)
38.
%20\%20+%20\%2042z%20&=&%2028yz%20\%20+%20\%2042z%5e%7b2%7d%20\%20+%20\%2042z\\\vspace*%7b9%20\hspace*%7b-10%7d%7d\\&=&%2042z%5e%7b2%7d%20\%20+%20\%2028yz%20\%20+%20\%2042z\end%7beqnarray%7d)
39.
%5e%7b2%7d%20\%20+%20\%204x%20&=&%20\frac%7bx%7d%7b8%7d\left(x%20\%20+%20\%208\right)\left(x%20\%20+%20\%208\right)%20\%20+%20\%204x\\\vspace*%7b9%20\hspace*%7b-10%7d%7d\\&=&%20\frac%7bx%7d%7b8%7d\left(x%5e%7b2%7d%20\%20+%20\%2016x%20\%20+%20\%2064)
40.
%20&=&%20\frac%7b9%7d%7ba%7d\left(9a%5e%7b6%7d%20\%20-%20\%2012ab\right)\\\vspace*%7b9%20\hspace*%7b-10%7d%7d\\&=&%20\frac%7b9\left(9a%5e%7b6%7d%20\%20-%20\%2012ab\right)%7d%7ba%7d\\\vspace*%7b9%20\hs)