1. Multiplying, Dividing, and Simplifying Radicals
10.3
Multiplying, Dividing, and Simplifying Radicals
1. Multiply radical expressions.
2. Divide radical expressions.
3. Use the product rule to simplify radical expressions.
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2. Product Rule for Radicals If and are both real numbers, then
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3. Example
Find the product and write the answer in simplest form. Assume all variables represent nonnegative values.
a. b.
Solution
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4. Raising an nth Root to the nth Power
For any nonnegative real number a,
Quotient Rule for Radicals
If and are both real numbers, then
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5. Example Simplify. Assume variables represent nonnegative values. a.
Solution b. a. b.
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6. 2. Use the product rule where a is a perfect nth power.
Simplifying nth Roots
1. Write the radicand as a product of the greatest possible perfect nth power and a number or expression that has no perfect nth power factors.
2. Use the product rule where a is a perfect nth power.
3. Find the nth root of the perfect nth power radicand.
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7. Example Simplify. a. b. Solution Solution
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8. Example Simplify. Solution
The greatest perfect square factor of 32x5 is 16x4.
Use the product rule of square roots to separate the factors into two radicals.
Find the square root of 16x4 and leave 2x in the radical.
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9. Example Simplify Solution
The greatest perfect square factor of 96a4b is 16a4.
Use the product rule of square roots to separate the factors into two radicals.
Find the square root of 16a4 and leave 6b in the radical.
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10. Simplify. Assume all variables represent nonnegative numbers.c) d)
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11. Simplify. Assume all variables represent nonnegative numbers.c) d)
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12. Simplify. a) b) c) d)
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13. Simplify. a) b) c) d)
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14. Adding, Subtracting, and Multiplying Radical Expressions
10.4
1. Add or subtract like radicals.
2. Use the distributive property in expressions containing radicals.
3. Simplify radical expressions that contain mixed operations.
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15. Like radicals: Radical expressions with identical radicands and identical indexes.
Adding Like Radicals
To add or subtract like radicals, add or subtract the coefficients and keep the radicals the same.
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16. Example Simplify. a. b. Solution a. b.
Combine the like radicals by subtracting the coefficients and keeping the radical. b.
Regroup terms.
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17. Example Simplify. a. b. Solution a. b. Factor 28. Simplify.
Combine like radicals. b.
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18. Example Find the product. Solution Use the distributive property.
Multiply.
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19. Example Find the product. Solution Use the distributive property.
Use the product rule.
Find the products.
Combine like radicals.
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20. Example Find the product. Solution Use (a – b)2 = a2 – 2ab – b2.
Simplify.
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21. Example Find the product. Solution Use (a + b)(a – b) = a2 – b2.
Simplify.
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22. Example Simplify. a. b. Solution b. a.
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23. Simplify. a) b) c) d)
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24. Simplify. a) b) c) d)
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25. Multiply. a) b) c) d)
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26. Multiply. a) b) c) d)
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27. Rationalizing Numerators and Denominators of Radical Expressions
10.5
1. Rationalize denominators.
2. Rationalize denominators that have a sum or difference with a square root term.
3. Rationalize numerators.
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28. Example Rationalize the denominator. Solution Multiply by Simplify.
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29. Example Rationalize the denominator. Solution
Use the quotient rule for square roots to separate the numerator and denominator into two radicals.
Solution
Multiply by
Simplify.
Warning: Never divide out factors common to a radicand and a number not under a radical.
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30. Rationalizing Denominators
To rationalize a denominator containing a single nth root, multiply the fraction by a 1 so that the product’s denominator has a radicand that is a perfect nth power.
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31. Example
Rationalize the denominator. Assume variables represent nonnegative values.
Solution
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32. Rationalizing a Denominator Containing a Sum or Difference
To rationalize a denominator containing a sum or difference with at least one square root term, multiply the fraction by a 1 whose numerator and denominator are the conjugate of the denominator.
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33. Example
Rationalize the denominator and simplify. Assume variables represent nonnegative values.
Solution
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34. Example
Rationalize the numerator. Assume variables represent nonnegative values.
Solution
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35. Rationalize the denominator.b) c) d)
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36. Rationalize the denominator.b) c) d)
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37. Rationalize the denominator.b) c) d)
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38. Rationalize the denominator.b) c) d)
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39. Radical Equations and Problem Solving
10.6
Radical Equations and Problem Solving
1. Use the power rule to solve radical equations.
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40. Radical equation: An equation containing at least one radical expression whose radicand has a variable.
Power Rule
If a = b, then an = bn.
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41. Example Solve. a. b. Solution a. b. Check Check True True
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42. Example Solve. Solution Check: The number 4 checks. The solution is 4.
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43. Example Solve. Solution Square both sides. Use FOIL.
Subtract x from both sides.
Subtract 7 from both sides.
Factor.
Use the zero-factor theorem.
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44. continued
Checks
True.
False.
Because 2 does not check, it is an extraneous solution. The only solution is 9.
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45. Example Solve. Solution Check
This solution does not check, so it is an extraneous solution. The equation has no solution; the solution set is
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46. Example Solve Solution Check The solution set is {13}.
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47. Solving Radical Equations To solve a radical equation:
1. Isolate the radical. (If there is more than one radical term, then isolate one of the radical terms.)
2. Raise both sides of the equation to the same power as the root index.
3. If all radicals have been eliminated, then solve. If a radical term remains, then isolate that radical term and raise both sides to the same power as its root index.
4. Check each solution. Any apparent solution that does not check is an extraneous solution.
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48. Solve. a) 6 b) 8 c) 9 d) no solution
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49. Solve. a) 6 b) 8 c) 9 d) no solution
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50. d) no real-number solution
Solve.
a) 2
b) 4 c)
d) no real-number solution
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51. d) no real-number solution
Solve.
a) 2
b) 4 c)
d) no real-number solution
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52. d) no real-number solution
Solve.
a) {-3, 4}
b) {-3}
c) {4}
d) no real-number solution
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53. d) no real-number solution
Solve.
a) {-3, 4}
b) {-3}
c) {4}
d) no real-number solution
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54. 10.7 Complex Numbers 1. Write imaginary numbers using i.
2. Perform arithmetic operations with complex numbers.
3. Raise i to powers.
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55. Imaginary unit: The number represented by i, where and i2 = 1.
Imaginary number: A number that can be expressed in the form bi, where b is a real number and i is the imaginary unit.
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56. Example
Write each imaginary number as a product of a real number and i.
a. b. c.
Solution
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57. Rewriting Imaginary Numbers
To write an imaginary number in terms of the imaginary unit i:
1. Separate the radical into two factors,
2. Replace with i.
3. Simplify
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58. Complex number: A number that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit.
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59. Example Add or subtract. (3 + 4i) – (4 – 12i) Solution
We subtract complex numbers just like we subtract polynomials.
(3 + 4i) – (4 – 12i) = (3 + 4i) + (4 + 12i)
= 7 + 16i
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60. Example Multiply. a. (8i)(4i) b. (9 – 4i)(3 + i) Solution
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61. Complex conjugates: The complex conjugate of a complex number a + bi is a – bi.
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62. Example Write in standard form. Solution Rationalize the denominator.
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63. Example Simplify. Solution = 1 Write i40 as (i4)10.
Replace i4 with 1.
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64. Simplify. (4 + 7i) – (2 + i) a) 2 + 7i2 b) 2 + 8i c) 6 + 6i
d) 6 + 8i
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65. Simplify. (4 + 7i) – (2 + i) a) 2 + 7i2 b) 2 + 8i c) 6 + 6i
d) 6 + 8i
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66. Multiply. (4 + 7i)(2 + i) a) 15 + 10i b) 1 + 10i c) 15 + 18i
d) i
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67. Multiply. (4 + 7i)(2 + i) a) 15 + 10i b) 1 + 10i c) 15 + 18i
d) i
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68. Write in standard form. a) b) c) d)
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69. Write in standard form. a) b) c) d)
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