1. Rational Exponents, Radicals, and Complex Numbers
Chapter 7
Rational Exponents, Radicals, and Complex Numbers

2. Radicals and Radical Functions
§ 7.1
Radicals and Radical Functions

3. Square Roots
Opposite of squaring a number is taking the square root of a number.
A number b is a square root of a number a if b2 = a.
In order to find a square root of a, you need a # that, when squared, equals a.

4. Principal Square Roots
Principal and Negative Square Roots
If a is a nonnegative number, then
is the principal or nonnegative square root of a
is the negative square root of a.

5. Radicands
Radical expression is an expression containing a radical sign.
Radicand is the expression under a radical sign.
Note that if the radicand of a square root is a negative number, the radical is NOT a real number.

6. Radicands
Example:

7. Perfect Squares
Square roots of perfect square radicands simplify to rational numbers (numbers that can be written as a quotient of integers).
Square roots of numbers that are not perfect squares (like 7, 10, etc.) are irrational numbers.
IF REQUESTED, you can find a decimal approximation for these irrational numbers.
Otherwise, leave them in radical form.

8. Perfect Square Roots
Radicands might also contain variables and powers of variables.
To avoid negative radicands, assume for this chapter that if a variable appears in the radicand, it represents positive numbers only.
Example:

9. Cube Roots
Cube Root
The cube root of a real number a is written as

10. Cube Roots
Example:

11. nth Roots Other roots can be found, as well.
The nth root of a is defined as
If the index, n, is even, the root is NOT a real number when a is negative.
If the index is odd, the root will be a real number.

12. nth Roots
Example:
Simplify the following.

13. nth Roots
Example:
Simplify the following. Assume that all variables represent positive numbers.

14. nth Roots
If the index of the root is even, then the notation represents a positive number.
But we may not know whether the variable a is a positive or negative value.
Since the positive square root must indeed be positive, we might have to use absolute value signs to guarantee the answer is positive.

15. Finding nth Roots If n is an even positive integer, then
If n is an odd positive integer, then

16. Finding nth Roots Simplify the following.
If we know for sure that the variables represent positive numbers, we can write our result without the absolute value sign.

17. Finding nth Roots Example: Simplify the following.
Since the index is odd, we don’t have to force the negative root to be a negative number.
If a or b is negative (and thus changes the sign of the answer), that’s okay.

18. Evaluating Rational Functions
We can also use function notation to represent rational functions.
For example,
Evaluating a rational function for a particular value involves replacing the value for the variable(s) involved.
Example:
Find the value

19. Root Functions
Since every value of x that is substituted into the equation
produces a unique value of y, the root relation actually represents a function.
The domain of the root function when the index is even, is all nonnegative numbers.
The domain of the root function when the index is odd, is the set of all real numbers.

20. Root Functions
We have previously worked with graphing basic forms of functions so that you have some familiarity with their general shape.
You should have a basic familiarity with root functions, as well.

21. Graphs of Root Functions
Example: x y
Graph
(6, )
x y
(4, 2)
(2, )
(1, 1) 6
(0, 0) 4 2 2 1

22. Graphs of Root Functions
Example:
Graph x y
x y 2 8 4
(8, 2)
(4, )
(1, 1) 1
(-1, -1)
(0, 0)
(-4, ) -1
(-8, -2) -4 -2 -8

23. § 7.2
Rational Exponents

24. Exponents with Rational Numbers
So far, we have only worked with integer exponents.
In this section, we extend exponents to rational numbers as a shorthand notation when using radicals.
The same rules for working with exponents will still apply.

25. Understanding a1/n Recall that a cube root is defined so that
However, if we let b = a1/3, then
Since both values of b give us the same a,
If n is a positive integer greater than 1 and is a real number, then

26. Using Radical Notation
Example:
Use radical notation to write the following. Simplify if possible.

27. Understanding am/n
If m and n are positive integers greater than 1 with m/n in lowest terms, then
as long as is a real number

28. Using Radical Notation
Example:
Use radical notation to write the following. Simplify if possible.

29. Understanding am/n
as long as a-m/n is a nonzero real number.

30. Using Radical Notation
Example:
Use radical notation to write the following. Simplify if possible.

31. Using Rules for Exponents
Example:
Use properties of exponents to simplify the following. Write results with only positive exponents.

32. Using Rational Exponents
Example:
Use rational exponents to write as a single radical.

33. Simplifying Radical Expressions
§ 7.3
Simplifying Radical Expressions

34. Product Rule for Radicals
If and are real numbers, then

35. Simplifying Radicals Example:
Simplify the following radical expressions.
No perfect square factor, so the radical is already simplified.

36. Simplifying Radicals Example:
Simplify the following radical expressions.

37. Quotient Rule Radicals
Quotient Rule for Radicals
If and are real numbers,

38. Simplifying Radicals Example:
Simplify the following radical expressions.

39. The Distance Formula Distance Formula
The distance d between two points (x1,y1) and (x2,y2) is given by

40. The Distance Formula Example:
Find the distance between (5, 8) and (2, 2).

41. The Midpoint Formula Midpoint Formula
The midpoint of the line segment whose endpoints are (x1,y1) and (x2,y2) is the point with coordinates

42. The Midpoint Formula Example:
Find the midpoint of the line segment that joins points P(5, 8) and P(2, 2).

43. Adding, Subtracting, and Multiplying Radical Expressions
§ 7.4
Adding, Subtracting, and Multiplying Radical Expressions

44. Sums and Differences
Rules in the previous section allowed us to split radicals that had a radicand which was a product or a quotient.
We can NOT split sums or differences.

45. Like Radicals
In previous chapters, we’ve discussed the concept of “like” terms.
These are terms with the same variables raised to the same powers.
They can be combined through addition and subtraction.
Similarly, we can work with the concept of “like” radicals to combine radicals with the same radicand.
Like radicals are radicals with the same index and the same radicand.
Like radicals can also be combined with addition or subtraction by using the distributive property.

46. Adding and Subtracting Radical Expressions
Example:
Can not simplify
Can not simplify

47. Adding and Subtracting Radical Expressions
Example:
Simplify the following radical expression.

48. Adding and Subtracting Radical Expressions
Example:
Simplify the following radical expression.

49. Adding and Subtracting Radical Expressions
Example:
Simplify the following radical expression. Assume that variables represent positive real numbers.

50. Multiplying and Dividing Radical Expressions
If and are real numbers,

51. Multiplying and Dividing Radical Expressions
Example:
Simplify the following radical expressions.

52. Rationalizing Numerators and Denominators of Radical Expressions
§ 7.5
Rationalizing Numerators and Denominators of Radical Expressions

53. Rationalizing the Denominator
Many times it is helpful to rewrite a radical quotient with the radical confined to ONLY the numerator.
If we rewrite the expression so that there is no radical in the denominator, it is called rationalizing the denominator.
This process involves multiplying the quotient by a form of 1 that will eliminate the radical in the denominator.

54. Rationalizing the Denominator
Example:
Rationalize the denominator.

55. Conjugates
Many rational quotients have a sum or difference of terms in a denominator, rather than a single radical.
In that case, we need to multiply by the conjugate of the numerator or denominator (which ever one we are rationalizing).
The conjugate uses the same terms, but the opposite operation (+ or ).

56. Rationalizing the Denominator
Example:
Rationalize the denominator.

57. Rationalizing the Numerator
An expression rewritten with no radical in the numerator is called rationalizing the numerator.
Example:

58. Rationalizing the Numerator
Example:
Rationalize the numerator.

59. Radical Equations and Problem Solving
§ 7.6
Radical Equations and Problem Solving

60. The Power Rule Power Rule
If both sides of an equation are raised to the same power, solutions of the new equation contain all the solutions of the original equation, but might also contain additional solutions.
A proposed solution of the new equation that is NOT a solution of the original equation is an extraneous solution.

61. Solving Radical Equations
Solving a Radical Equation
Isolate one radical on one side of the equation.
Raise each side of the equation to a power equal to the index of the radical and simplify.
If the equation still contains a radical term, repeat Steps 1 and 2. If not, solve the equation.
Check all proposed solutions in the original equation.

62. Solving Radical Equations
Example:
Solve the following radical equation.
Substitute into the original equation.
true
So the solution is x = 2.

63. Solving Radical Equations
Example:
Solve the following radical equation.

64. Solving Radical Equations
Example continued:
Substitute the value for x into the original equation, to check the solution.
true
So the solution is x = 3.
false

65. Solving Radical Equations
Example:
Solve the following radical equation.

66. Solving Radical Equations
Example continued:
Substitute the value for x into the original equation, to check the solution.
false
So the solution is .

67. Solving Radical Equations
Example:
Solve the following radical equation.

68. Solving Radical Equations
Example continued:
Substitute the value for x into the original equation, to check the solution.
true
true
So the solution is x = 4 or 20.

69. Solving Radical Equations
Example:
Solve the following radical equation.
Substitute into the original equation.
true
So the solution is x = 24.

70. Solving Radical Equations
Example:
Solve the following radical equation.
Substitute into the original equation.
Does NOT check, since the left side of the equation is asking for the principal square root.
So the solution is .

71. The Pythagorean Theorem
If a and b are the lengths of the legs of a right triangle and c is the length of the hypotenuse, then
a2 + b2 = c2
There are several applications in this section that require the use of the Pythagorean Theorem in order to solve.

72. Using the Pythagorean Theorem
Example:
Find the length of the hypotenuse of a right triangle when the length of the two legs are 2 inches and 7 inches.
c2 = = = 53
c = inches

73. § 7.7
Complex Numbers

74. Imaginary Numbers
Previously, when we encountered square roots of negative numbers in solving equations, we would say “no real solution” or “not a real number”.
Imaginary Unit
The imaginary unit i, is the number whose square is – 1. That is,

75. The Imaginary Unit, i Example:
Write the following with the i notation. i i i i

76. Complex Numbers
Real numbers and imaginary numbers are both subsets of a new set of numbers.
Complex Numbers
A complex number is a number that can be written in the form a + bi, where a and b are real numbers.

77. Standard Form of Complex Numbers
Complex numbers can be written in the form a + bi (called standard form), with both a and b as real numbers.
a is a real number and bi would be an imaginary number.
If b = 0, a + bi is a real number.
If a = 0, a + bi is an imaginary number.

78. Standard Form of Complex Numbers
Example:
Write each of the following in the form of a complex number in standard form a + bi.
6 =
6 + 0i
8i =
0 + 8i i i
6 + 5i

79. Adding and Subtracting Complex Numbers
Sum or Difference of Complex Numbers
If a + bi and c + di are complex numbers, then their sum is
(a + bi) + (c + di) = (a + c) + (b + d)i
Their difference is
(a + bi) – (c + di) = (a – c) + (b – d)i

80. Adding and Subtracting Complex Numbers
Example:
Add or subtract the following complex numbers. Write the answer in standard form a + bi.
(4 + 6i) + (3 – 2i) =
(4 + 3) + (6 – 2)i =
7 + 4i
(8 + 2i) – (4i) =
(8 – 0) + (2 – 4)i =
8 – 2i

81. Multiplying Complex Numbers
The technique for multiplying complex numbers varies depending on whether the numbers are written as single term (either the real or imaginary component is missing) or two terms.

82. Multiplying Complex Numbers
Note that the product rule for radicals does NOT apply for imaginary numbers.

83. Multiplying Complex Numbers
Example:
Multiply the following complex numbers.
8i · 7i
56i2
56(1)
56

84. Multiplying Complex Numbers
Example:
Multiply the following complex numbers. Write the answer in standard form a + bi.
5i(4 – 7i)
20i – 35i2
20i – 35(–1)
20i + 35 i

85. Multiplying Complex Numbers
Example:
Multiply the following complex numbers. Write the answer in standard form a + bi.
(6 – 3i)(7 + 4i)
i – 21i – 12i2
42 + 3i – 12(–1)
42 + 3i + 12
54 + 3i

86. Complex Conjugate Complex Conjugates
The complex numbers (a + bi) and (a – bi) are complex conjugates of each other, and
(a + bi)(a – bi) = a2 + b2

87. Complex Conjugate The conjugate of a + bi is a – bi.
The product of (a + bi) and (a – bi) is
(a + bi)(a – bi)
a2 – abi + abi – b2i2
a2 – b2(–1)
a2 + b2, which is a real number.

88. Dividing Complex Numbers
Example:
Use complex conjugates to divide the following complex numbers. Write the answer in standard form.

89. Dividing Complex Numbers
Example:
Divide the following complex numbers.

90. Patterns of i
The powers recycle through each multiple of 4.

91. Patterns of i
Example:
Simplify each of the following powers.