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# Intermediate Algebra

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### Intermediate Algebra

1. 1. E xam 4 M aterial Radicals, Rational E xponents & E quations
2. 2. Square Roots  A square root of a real number “ a” is a real number that multiplies by itself to give “ a” 3 What is a square root of 9? 3 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? N real number multiplied by itself can give a negative answer o  Every positive real number “ a” has two square roots that have equal absolute values, but opposite signs 16 T two square roots of 16 are: he 16 and simplified : 4 and  4 5 and  5 T two square roots of 5 are: he (Positive Square Root : PRINCIPLE ROOT)
3. 3. Even Roots (2,4,6,…)  The even “ n th ” 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 exist 4  16 does not negative number  Every positive real number “ a” has two even roots that have equal absolute values, but opposite signs T fourth roots of 16: he 4 16 and  4 16 simplified : 2 and  2 T fourth roots of 7: he 4 4 7 and - 7 (Positive Even Root : PRINCIPLE ROOT)
4. 4. Radical Expressions  On the previous slides we have used symbols of the form: n a  This is called a radical expression and the parts of the expression are named: Index: Radical S ign : n Radicand:  E xample: a 5 8 Index : 5 Radicand : 8
5. 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: 3 8   2 3 6 No such thing as a principle cube root!
6. 6. Odd Roots (3,5,7,…)  The odd n th 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 T only fifth root of - 32: he T only fifth root of -7: he 3  32   2 5 7
7. 7. Rational, Irrational, and Non-real Radical Expressions n a  is non-real only if the radicand is negative and the index is even 6  20 is non - real because radicand is negative and index is even n  arepresents a rational number only if the radicand can be written as a “ perfect n th ” power of an integer or the ratio of two integers  32 is rational because  32    2  5 5  32  2 5  n a represents an irrational number only if it is a real number and the radicand can not be written as “ perfect n th ” power of an integer or the ratio of two integers . 4 8 is irrational because 8 is not the fourth power 4 8 of an integer or the ratio of two integers
8. 8. Homework Problems  S ection: 10.1  Page: 666  Problems: A ll: 1 – 6, Odd: 7 – 31, 39 – 57, 65 – 91  M yM athL ab Homework A ssignment 10.1 for practice  M yM athL ab Quiz 10.1 for grade
9. 9. Exponential Expressions an “ a” is called the base “ n” is called the exponent  If “ n” is a natural number then “ a n ” means that “ a” is to be multiplied by itself “ n” times. E xample: What is the value of 2 4 ? (2)(2)(2)(2) = 16  A n exponent applies only to the base (what it touches) E xample: What is the value of: - 3 4 ? - (3)(3)(3)(3) = - 81 E xample: What is the value of: (- 3)4 ? (- 3)(- 3)(- 3)(- 3) = 81  M eanings of exponents that are not natural numbers will be discussed in this unit.
10. 10. Negative Exponents: a -n  A negative exponent has the meaning: “ reciprocate the base and make the exponent positive” n n 1 a   a Examples: 2 2 1 1 3     3 9 3 3 2  3  27      3 2 8 .
11. 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 am  a mn an E xamples: 54  547  53 57 x12  x12  4  x 8 . x4
12. 12. Rational Exponents (a1/n) and Roots 1 n  A n exponent of the form has the meaning: “ the n th root of the base, if it exists, and, if there are two nth roots, it means the principle (positive) one” 1 th a , if it exists, is the n root of a n 1 (If there are two n th roots, a is the principle (positive) one) n 1 ( a multiplies by itself n times to give a) n
13. 13. Examples of Rational Exponent of the Form: 1/n 1 100 2  10 (positive square root of 100) 1 5  5 (positive square root of 5) 2 1   3 2  (Does not exist! ) 1  3   3 (negative square root of 3) 2 1 7  4 4 7 (positive fourth root of 7) 1   9 7  7  9 (seventh root of negative 9) . 1   8 6  (Does not exist! )
14. 14. Summary Comments about Meaning of a1/n  When n is odd:  a 1/n always exists and is either positive, negative or zero depending on whether “ a” is positive, negative or zero  When n is even:  a 1/n never exists when “a” is negative  a 1/n always exists and is positive or zero depending on whether “ a” is positive or zero
15. 15. Rational Exponents of the Form: m/n  A n exponent of the form m/ has two equivalent n meanings: (1) a m/n means find the n th root of “ a” , then raise it to the power of “ m” (assuming that the n th root of “ a” exists) (2) a m/n means raise “ a” to the power of “ m” then take the n th root of a m (assuming that the n th root of “ a m” exists)
16. 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: 82/3 = 4 (8 squared is 64, and the cube root of 64 is 4)
17. 17. Definitions and Rules for Exponents  A ll the rules learned for natural number exponents continue to be true for both positive and negative rational exponents: Product Rule: a ma n = a m+n 4 2 6 37  37  37 2 Quotient Rule: a m/ n = a m-n a 3 7  2 4  3 7 Negative Exponents: a -n = (1/ n a) 3 7 4 4  1 7 3 7    3 .
18. 18. Definitions and Rules for Exponents   4 2 7 8 3   3 7 49 Power Rules: (a ) = a m n mn     2 2 2 (ab) = a b m m m  3x  7  3 x 7 7 2 2 (a/ m = a m /b m b) 3 3 7 7    2 4 47 0 Zero Exponent: a 0 = 1 (a not zero) 3    1 . 4
19. 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: a mb  n cr d s r s  m n c d a b  This rule applies to all types of exponents  Often used to make all exponents positive
20. 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  S implify 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. 21. Simplify the Expression:   1 8 y y  3 2 8     8y  3 16 3 1 7 1  2 1 y 12 39 2 y y 4 6  2 y 12 8 y y 23 21  8 3 1 7 8  2 1 y y 4 6 y y 12 3 2 6   8y y 3 3 16 9 2 7 32  2 1 y y 12 12 y y 12 12
22. 22. Applying Rules of Exponents in Multiplying and Factoring  Multiply:  1  1   1  1 1  1  1 1  1  x 2  2  x 2  x 2   x 2 x 2  x 2 x 2  2 x 2  2 x 2       1 1 1 1 1 1 1 1       x 2 2 x 2 2  2x  2x 2 2  x 0  x 1  2 x  2 x 2 2 1 1   1  x 1  2 x  2 x 2 2  Factor out the indicated factor:  3 1  3 5x 4 x ;x 4 4 3  3 4   3  5  x  x 4  __  __  4 4   x 5  x   x 4  
23. 23. Radical Notation  Roots of real numbers may be indicated by means of either rational exponent notation or radical notation: n a is called a RADICAL (expressio n) is called a RADICAL SIGN n is called the INDEX a is called the RADICAND
24. 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. 25. Converting Between Radical and Rational Exponent Notation  A n 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: m 2 a  a n n m 83  3 82  3 64  4 1.  a  8 m 2   2  4 m 2 2 2. a  n n 8  3 3  These definitions assume that the nth root of “ a” exists
26. 26. Examples  5 4 4 5  7 7 OR 7 5 4 9 5 8  8 9 5   3 3 4x  4 x11 11 3 OR 4 x 11 .
27. 27. n n x .  If “ n” is even, then this notation means principle (positive) root: n x  x n (absolute value needed to insure positive answer)  If “ n” is odd, then: x x n n  If we assume that “ x” is positive (which we often do) then we can say that: . n x xn
28. 28. Homework Problems  S ection: 10.2  Page: 675  Problems: A ll: 1 – 10, Odd: 11 – 47, 51 – 97  M yM athL ab Homework A ssignment 10.2 for practice  M yM athL ab Quiz 10.2 for grade
29. 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: n a b  ab n n 4 3 5  3  5  15 4 4 4 n ab  a bn n 3 16  8 2  2 2 3 3 3
30. 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 4 n a n a 5 5   4 n b b 4 7 7  This rule works both directions: a na 5 35 3 5 n n 3 3  . b b 8 8 2
31. 31. Root of a Root Rule for Radicals  When you take the m th root of the n th root of a radicand “ a” , it is the same as taking a single root of “ a” using an index of “ mn” m n a  mn a 4 3 6 12 6 .
32. 32. NO Similar Rules for Sum and Difference of Radicals n a n b  n ab . 3 27  8  35 3 3 3  2  35 3 n a n b  n a b 3 27  3 8  3 19 3  2  19 3
33. 33. Simplifying Radicals  A radical must be simplified if any of the following conditions exist: 2. S ome factor of the radicand has an exponent that is bigger than or equal to the index 3. There is a radical in a denominator (denominator needs to be “ rationalized” ) 4. The radicand is a fraction 5. All of the factors of the radicand have exponents that share a common factor with the index
34. 34. Simplifying when Radicand has 3 4 2 Exponent Too Big 1. 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 2. S implify the first radical 3 33 2 2 3 2 2
35. 35. Problem? Example 2 5 3 24 x y Is there another exponent t hat is too big? 3 2 5 3 2 3x y Write this as a product of two radicals : 3 33 2 2 3 2 y 3x y Simplify the first radical : 2 2 2 y 3x y 3
36. 36. Simplifying when a Denominator Contains a Single Radical of Index “n” 1. S implify the top and bottom separately to get rid of exponents under the radical that are too big 2. M ultiply the whole fraction by a special kind of “ 1” where 1 is in the form of: n m n m and m is the product of all the factors required to make every exponent in the radicand be equal to quot;nquot; 7. S implify to eliminate the radical in the denominator
37. 37. Example 3 3 3 3    3 6 2 3 6 55 2 3 5 4x y 5 2 x y 5 y 2 x y y 5 22 x 3 y 3 5 23 x 2 y 4 35 23 x 2 y 4 35 23 x 2 y 4     y5 22 x3 y 5 23 x 2 y 4 y 5 25 x 5 y 5 2 xy 2 2 4 3 8x y 5  2 2 xy
38. 38. Simplifying when Radicand is a Fraction 1. Use the quotient rule to write the single radical as a quotient of two radicals 2. Use the rules already learned for simplifying when there is a radical in a denominator
39. 39. Example 3 3 5 5 3 5 3 5 23 5 3 23 5  5     4 4 5 22 5 2 2 5 2 3 5 25 5 24  2
40. 40. Simplifying when All Exponents in Radicand Share a Common Factor with Index 1. Divide out the common factor from the index and all exponents 4 6 8 2 6 23 x y All exponents in radicand and index share what factor? 2 Dividing all exponents in and index by 2 gives : 3 2 3 4 23 x y  3 x 3 3 33 2 2 xy  3 x 3 4 xy Problem?
41. 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. 42. Example 4 ab ab 34 a 2b 4 b 4 b 4 b 4 a3 4 3 3  4 4 4 4 a 3b 3 ab a a a 4 a3 4 3 4 ab 3 ab  4 a4  a
43. 43. Right Triangle  A “ right triangle” is a triangle that has a 90 0 angle (where two sides intersect perpendicularly) b c  hypotenuse  90 0  The side opposite the right angle is called the a “ hypotenuse” and is traditionally identified as side “ c”  The other two sides are called “ legs” and are traditionally labeled “ a” and “ b”
44. 44. Pythagorean Theorem  In a right triangle, the square of the hypotenuse is always equal to the sum of the squares of the legs: c  a b 2 2 2 c b 90 0 a
45. 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. 5  Note that Pythagorean Theorem 3 true: is 90 0 4 c  a b 2 2 2 5  4 3 2 2 2 25  16  9
46. 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. 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”  S quare root both sides to find the desired answer
48. 48. Example  Given a right triangle with a  7 and c  25 find the other side. c  a b 2 2 2 25  7  b 2 2 2 625  49  b 2 625  49  49  49  b 2 576  b 2 24  576  b
49. 49. Homework Problems  S ection: 10.3  Page: 685  Problems: Odd: 7 – 19, 23 – 57, 61 – 107  M yM athL ab Homework A ssignment 10.3 for practice  M yM athL ab Quiz 10.3 for grade
50. 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 4 4 3 Which are like radicals? 3 5 , 4 5 , - 2 5 and 3 5  When “like radicals” are added or subtracted, the result is a “like radical” with coefficient equal to the sum or difference of the coefficients 3 5 2 5  54 5 4 4 -2 5  3 5  4 3 Okay as is - can' t combine unlike radicals
51. 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. 52. Example(yet) Not like terms Simplify individual radicals : 3 128  5 3 2  2 3 16  3 27  5 3 2  2 3 2 4  3 23 23 3 2  5 3 2  2 3 23 3 2  2  23 2  5 3 2  2  2 3 2 All like radicals :  43 2  5 3 2  4 3 2 3 2 3
53. 53. Homework Problems  S ection: 10.4  Page: 691  Problems: Odd: 5 – 57  M yM athL ab Homework A ssignment 10.4 for practice  M yM athL ab Quiz 10.4 for grade
54. 54. Simplifying when there is a Single Radical Term in a Denominator 1. Simplify the radical in the denominator 2. If the denominator still contains a radical, multiply the fraction by “ 1” where “ 1” is in the form of a “special radical” over itself 3. The “ special radical” is one that contains the factors necessary to make the denominator radical factors have exponents equal to index 4. Simplify radical in denominator to eliminate it
55. 55. Example 2 3 Simplify denominato r : 3 9x 1 3 2 3 2 3 2 3 x Multiply by special quot;1quot;: 6x 3 2 3 3x 2 Use product rule : 3x 3 2 3 2 3 x 3x 3 2  3x 2 Simplify denominato r : 3 3 3 3 x
56. 56. Simplifying to Get Rid of a Binomial Denominator that Contains One or Two Square Root Radicals 1. Simplify the radical(s) in the denominator 2. 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 3. The “ special binomial radical” is the conjugate of the denominator (same terms – opposite sign) 4. Complete multiplication (the denominator will contain no radical)
57. 57. Example Radical in denominator doesn' t need simplifying 5 3 2 Multiply fraction by special one : 5 3 2 Distribute on top :  3 2 3 2 FOIL on bottom : 15  10 9 4 Simplify bottom : 15  10 3 2 15  10
58. 58. Homework Problems  S ection: 10.5  Page: 700  Problems: Odd: 7 – 105  M yM athL ab Homework A ssignment 10.5 for practice  M yM athL ab Quiz 10.5 for grade
59. 59. Radical Equations  A n equation is called a radical equation if it contains a variable in a radicand  E xamples: x x3  5 x  x 5 1 3 x  4  3 2x  0
60. 60. Solving Radical Equations 1. Isolate ONE radical on one side of the equal sign 2. Raise both sides of equation to power necessary to eliminate the isolated radical 3. Solve the resulting equation to find “ apparent solutions” 4. 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. 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:  2  2 , however :  - 2    2  ,  - 2    2  , etc. 2 2 4 4  Raising both sides to an odd power never makes a false statement true:  2  2 , and :  - 2    2  ,  - 2    2  , etc. . 3 3 5 5
62. 62. Example of Solving Radical Equation Check x  4 x x3  5 4 43  5? x5  x3 4 1 5?  x  5 2   x3  2 35 x  4 is NOT a solution x  10 x  25  x  3 2 Check x  7 x 2  11x  28  0 7 73  5?  x  4 x  7   0 7 4 5? x  4  0 OR x  7  0 55 x  4 OR x  7 x7 IS a solution
63. 63. Example of Solving Radical Equation x  x 5 1 Check x  4 x  5  1 x 4  4  5 1?    2 x  5  1 x  2 4  9 1? x  5  1 2 x  x 2  3 1? 4  2 x 5 1 x  4 is NOT a solution 2 x     2  x 2 2 Equation has No Solution! 4x 
64. 64. Example of Solving Radical Equation 3 x  4  3 2x  0 3 x  4  2x 3  3 x4   3 3 2x  3 x  4  2x 4x (No need to check)
65. 65. Homework Problems  S ection: 10.6  Page: 709  Problems: Odd: 7 – 57  M yM athL ab Homework A ssignment 10.6 for practice  M yM athL ab Quiz 10.6 for grade