1. Exam 4 Material Radicals, Rational Exponents & Equations
Intermediate Algebra
Exam 4 Material
Radicals, Rational Exponents & Equations

2. Square Roots
A square root of a real number “a” is a real number that multiplies by itself to give “a”
What is a square root of 9?
What is another square root of 9?
What is the square root of -4 ?
Square root of – 4 does not exist in the real number system
Why is it that square roots of negative numbers do not exist in the real number system?
No real number multiplied by itself can give a negative answer
Every positive real number “a” has two square roots that have equal absolute values, but opposite signs
The two square roots of 16 are:
The two square roots of 5 are:

3. Even Roots (2,4,6,…)
The even “nth” root of a real number “a” is a real number that multiplies by itself “n” times to give “a”
Even roots of negative numbers do not exist in the real number system, because no real number multiplied by itself an even number of times can give a negative number
Every positive real number “a” has two even roots that have equal absolute values, but opposite signs
The fourth roots of 16:
The fourth roots of 7:

4. Radical Expressions
On the previous slides we have used symbols of the form:
This is called a radical expression and the parts of the expression are named:
Index:
Radical Sign :
Radicand:
Example:

5. Cube Roots
The cube root of a real number “a” is a real number that multiplies by itself 3 times to give “a”
Every real number “a” has exactly one cube root that is positive when “a” is positive, and negative when “a” is negative
Only cube root of – 8:
Only cube root of 6:

6. Odd Roots (3,5,7,…)
The odd nth root of a real number “a” is a real number that multiplies by itself “n” times to give “a”
Every real number “a” has exactly one odd root that is positive when “a” is positive, and negative when “a” is negative
The only fifth root of - 32:
The only fifth root of -7:

7. Rational, Irrational, and Non-real Radical Expressions
is non-real only if the radicand is negative and the index is even
represents a rational number only if the radicand can be written as a “perfect nth” power of an integer or the ratio of two integers
represents an irrational number only if it is a real number and the radicand can not be written as “perfect nth” power of an integer or the ratio of two integers .

8. Homework Problems Section: 10.1 Page: 666
Problems: All: 1 – 6, Odd: 7 – 31, – 57, 65 – 91
MyMathLab Homework Assignment 10.1 for practice
MyMathLab Quiz 10.1 for grade

9. Exponential Expressionsan
“a” is called the base
“n” is called the exponent
If “n” is a natural number then “an” means that “a” is to be multiplied by itself “n” times.
Example: What is the value of 24 ?
(2)(2)(2)(2) = 16
An exponent applies only to the base (what it touches)
Example: What is the value of: ?
- (3)(3)(3)(3) = - 81
Example: What is the value of: (- 3)4 ?
(- 3)(- 3)(- 3)(- 3) = 81
Meanings of exponents that are not natural numbers will be discussed in this unit.

10. Negative Exponents: a-n
A negative exponent has the meaning: “reciprocate the base and make the exponent positive”
Examples: .

11. Quotient Rule for Exponential Expressions
When exponential expressions with the same base are divided, the result is an exponential expression with the same base and an exponent equal to the numerator exponent minus the denominator exponent
Examples: .

12. Rational Exponents (a1/n) and Roots
An exponent of the form
has the meaning: “the nth root of the base, if it exists, and, if there are two nth roots, it means the principle (positive) one”

13. Examples of Rational Exponent of the Form: 1/n.

14. Summary Comments about Meaning of a1/n
When n is odd:
a1/n always exists and is either positive, negative or zero depending on whether “a” is positive, negative or zero
When n is even:
a1/n never exists when “a” is negative
a1/n always exists and is positive or zero depending on whether “a” is positive or zero

15. Rational Exponents of the Form: m/n
An exponent of the form m/n has two equivalent meanings:
(1) am/n means find the nth root of “a”, then raise it to the power of “m”
(assuming that the nth root of “a” exists)
(2) am/n means raise “a” to the power of “m” then take the nth root of am
(assuming that the nth root of “am” exists)

16. Example of Rational Exponent of the Form: m/n
82/3
by definition number 1 this means find the cube root of 8, then square it:
82/3 = 4
(cube root of 8 is 2, and 2 squared is 4)
by definition number 2 this means raise 8 to the power of 2 and then cube root that answer:
(8 squared is 64, and the cube root of 64 is 4)

17. Definitions and Rules for Exponents
All the rules learned for natural number exponents continue to be true for both positive and negative rational exponents:
Product Rule: aman = am+n
Quotient Rule: am/an = am-n
Negative Exponents: a-n = (1/a)n .

18. Definitions and Rules for Exponents
Power Rules: (am)n = amn
(ab)m = ambm
(a/b)m = am / bm
Zero Exponent: a0 = 1 (a not zero) .

19. “Slide Rule” for Exponential Expressions
When both the numerator and denominator of a fraction are factored then any factor may slide from the top to bottom, or vice versa, by changing the sign on the exponent
Example: Use rule to slide all factors to other part of the fraction:
This rule applies to all types of exponents
Often used to make all exponents positive

20. Simplifying Products and Quotients Having Factors with Rational Exponents
All factors containing a common base can be combined using rules of exponents in such a way that all exponents are positive:
Use rules of exponents to get rid of parentheses
Simplify top and bottom separately by using product rules
Use slide rule to move all factors containing a common base to the same part of the fraction
If any exponents are negative make a final application of the slide rule

21. Simplify the Expression:

22. Applying Rules of Exponents in Multiplying and Factoring
Factor out the indicated factor:

23. Radical Notation
Roots of real numbers may be indicated by means of either rational exponent notation or radical notation:

24. Notes About Radical Notation
If no index is shown it is assumed to be 2
When index is 2, the radical is called a “square root”
When index is 3, the radical is called a “cube root”
When index is n, the radical is called an “nth root”
In the real number system, we can only find even roots of non-negative radicands. There are always two roots when the index is even, but a radical with an even index always means the positive (principle) root
We can always find an odd root of any real number and the result is positive or negative depending on whether the radicand is positive or negative

25. Converting Between Radical and Rational Exponent Notation
An exponential expression with exponent of the form “m/n” can be converted to radical notation with index of “n”, and vice versa, by either of the following formulas: 1. 2.
These definitions assume that the nth root of “a” exists

26. Examples .

27. . If “n” is even, then this notation means principle (positive) root:
If “n” is odd, then:
If we assume that “x” is positive (which we often do) then we can say that: .

28. Homework Problems Section: 10.2 Page: 675
Problems: All: 1 – 10, Odd: 11 – 47, – 97
MyMathLab Homework Assignment 10.2 for practice
MyMathLab Quiz 10.2 for grade

29. Product Rule for Radicals
When two radicals are multiplied that have the same index they may be combined as a single radical having that index and radicand equal to the product of the two radicands:
This rule works both directions:

30. Quotient Rule for Radicals
When two radicals are divided that have the same index they may be combined as a single radical having that index and radicand equal to the quotient of the two radicands
This rule works both directions: .

31. Root of a Root Rule for Radicals
When you take the mth root of the nth root of a radicand “a”, it is the same as taking a single root of “a” using an index of “mn” .

32. NO Similar Rules for Sum and Difference of Radicals.

33. Simplifying Radicals
A radical must be simplified if any of the following conditions exist:
Some factor of the radicand has an exponent that is bigger than or equal to the index
There is a radical in a denominator (denominator needs to be “rationalized”)
The radicand is a fraction
All of the factors of the radicand have exponents that share a common factor with the index

34. Simplifying when Radicand has Exponent Too Big
Use the product rule to write the single radical as a product of two radicals where the first radicand contains all factors whose exponents match the index and the second radicand contains all other factors
Simplify the first radical

35. Example

36. Simplifying when a Denominator Contains a Single Radical of Index “n”
Simplify the top and bottom separately to get rid of exponents under the radical that are too big
Multiply the whole fraction by a special kind of “1” where 1 is in the form of:
Simplify to eliminate the radical in the denominator

37. Example

38. Simplifying when Radicand is a Fraction
Use the quotient rule to write the single radical as a quotient of two radicals
Use the rules already learned for simplifying when there is a radical in a denominator

39. Example

40. Simplifying when All Exponents in Radicand Share a Common Factor with Index
Divide out the common factor from the index and all exponents

41. Simplifying Expressions Involving Products and/or Quotients of Radicals with the Same Index
Use the product and quotient rules to combine everything under a single radical
Simplify the single radical by procedures previously discussed

42. Example

43. Right Triangle
A “right triangle” is a triangle that has a 900 angle (where two sides intersect perpendicularly)
The side opposite the right angle is called the “hypotenuse” and is traditionally identified as side “c”
The other two sides are called “legs” and are traditionally labeled “a” and “b”

44. Pythagorean Theorem
In a right triangle, the square of the hypotenuse is always equal to the sum of the squares of the legs:

45. Pythagorean Theorem Example
It is a known fact that a triangle having shorter sides of lengths 3 and 4, and a longer side of length 5, is a right triangle with hypotenuse 5.
Note that Pythagorean Theorem is true:

46. Using the Pythagorean Theorem
We can use the Pythagorean Theorem to find the third side of a right triangle, when the other two sides are known, by finding, or estimating, the square root of a number

47. Using the Pythagorean Theorem
Given two sides of a right triangle with one side unknown:
Plug two known values and one unknown value into Pythagorean Theorem
Use addition or subtraction to isolate the “variable squared”
Square root both sides to find the desired answer

48. Example
Given a right triangle with find the other side.

49. Homework Problems Section: 10.3 Page: 685
Problems: Odd: 7 – 19, 23 – 57, – 107
MyMathLab Homework Assignment 10.3 for practice
MyMathLab Quiz 10.3 for grade

50. Adding and Subtracting Radicals
Addition and subtraction of radicals can always be indicated, but can be simplified into a single radical only when the radicals are “like radicals”
“Like Radicals” are radicals that have exactly the same index and radicand, but may have different coefficients
Which are like radicals?
When “like radicals” are added or subtracted, the result is a “like radical” with coefficient equal to the sum or difference of the coefficients

51. Note Concerning Adding and Subtracting Radicals
When addition or subtraction of radicals is indicated you must first simplify all radicals because some radicals that do not appear to be like radicals become like radicals when simplified

52. Example

53. Homework Problems Section: 10.4 Page: 691 Problems: Odd: 5 – 57
MyMathLab Homework Assignment 10.4 for practice
MyMathLab Quiz 10.4 for grade

54. Simplifying when there is a Single Radical Term in a Denominator
Simplify the radical in the denominator
If the denominator still contains a radical, multiply the fraction by “1” where “1” is in the form of a “special radical” over itself
The “special radical” is one that contains the factors necessary to make the denominator radical factors have exponents equal to index
Simplify radical in denominator to eliminate it

55. Example

56. Simplifying to Get Rid of a Binomial Denominator that Contains One or Two Square Root Radicals
Simplify the radical(s) in the denominator
If the denominator still contains a radical, multiply the fraction by “1” where “1” is in the form of a “special binomial radical” over itself
The “special binomial radical” is the conjugate of the denominator (same terms – opposite sign)
Complete multiplication (the denominator will contain no radical)

57. Example

58. Homework Problems Section: 10.5 Page: 700 Problems: Odd: 7 – 105
MyMathLab Homework Assignment 10.5 for practice
MyMathLab Quiz 10.5 for grade

59. Radical Equations
An equation is called a radical equation if it contains a variable in a radicand
Examples:

60. Solving Radical Equations
Isolate ONE radical on one side of the equal sign
Raise both sides of equation to power necessary to eliminate the isolated radical
Solve the resulting equation to find “apparent solutions”
Apparent solutions will be actual solutions if both sides of equation were raised to an odd power, BUT if both sides of equation were raised to an even power, apparent solutions MUST be checked to see if they are actual solutions

61. Why Check When Both Sides are Raised to an Even Power?
Raising both sides of an equation to a power does not always result in equivalent equations
If both sides of equation are raised to an odd power, then resulting equations are equivalent
If both sides of equation are raised to an even power, then resulting equations are not equivalent (“extraneous solutions” may be introduced)
Raising both sides to an even power, may make a false statement true:
Raising both sides to an odd power never makes a false statement true: .

62. Example of Solving Radical Equation

63. Example of Solving Radical Equation

64. Example of Solving Radical Equation

65. Homework Problems Section: 10.6 Page: 709 Problems: Odd: 7 – 57
MyMathLab Homework Assignment 10.6 for practice
MyMathLab Quiz 10.6 for grade