1. Math 083 Bianco Warm Up! List all the perfect squares you know.
List all the perfect cubes you know

2. What is the difference?

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

4. Radicals and Radical Functions
§ 7.1
Radicals and Radical Functions

5. 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.

6. 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.

7. 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.

8. Radicands
Example:

9. 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.

10. 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:

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

12. Cube Roots
Example:

13. 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.

14. nth Roots
Example:
Simplify the following.

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

16. 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.

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

18. 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.

19. 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.

20. 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

21. 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.

22. 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.

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

24. 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

25. § 7.2
Rational Exponents

26. 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.

27. 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

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

29. 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

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

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

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

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

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

35. Simplifying Radical Expressions
§ 7.3
Simplifying Radical Expressions

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

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

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

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

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

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

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

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

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

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

53. 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.

54. 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.

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

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

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

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

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

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

65. Homework
7.1 #’s
7.2 #’s
7.3 #’s
7.4 #’s
Tuesday 7.6/7.7