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Copyright © 2011 Pearson Education, Inc.
Rational Exponents, Radicals, and Complex Numbers
CHAPTER
10.1 Radical Expressions and Functions10.2 Rational Exponents10.3 Multiplying, Dividing, and Simplifying Radicals10.4 Adding, Subtracting, and Multiplying Radical
Expressions10.5 Rationalizing Numerators and Denominators of
Radical Expressions10.6 Radical Equations and Problem Solving 10.7 Complex Numbers
1100
Copyright © 2011 Pearson Education, Inc.
Radical Expressions and Functions10.110.1
1. Find the nth root of a number.2. Approximate roots using a calculator.3. Simplify radical expressions.4. Evaluate radical functions.5. Find the domain of radical functions.6. Solve applications involving radical functions.
Slide 10- 3Copyright © 2011 Pearson Education, Inc.
Evaluating nth rootsWhen evaluating a radical expression , the sign of a
and the index n will determine possible outcomes.If a is nonnegative, then , where and bn = a.If a is negative and n is even, then there is no real-number root.If a is negative and n is odd, then , where b is negative and bn = a.
n a
n a b 0b
n a b
nth root: The number b is an nth root of a number a if bn = a.
Slide 10- 4Copyright © 2011 Pearson Education, Inc.
Example 1
Evaluate each root, if possible.a.
b.
c.
169
Solution 169 13
0.49
Solution 0.49 0.7
100
Solution 100 is not a real number because there is no real number whose square is –100.
Slide 10- 5Copyright © 2011 Pearson Education, Inc.
continued
Evaluate each root, if possible.d.
e.
f.
144
Solution 144 12
49
144
Solution 49
144
49
144
7
12
3 27
Solution 3 27 3
Slide 10- 6Copyright © 2011 Pearson Education, Inc.
continued
Evaluate each root, if possible.g.
h.
3 27
Solution 3 27
4 81
Solution 4 81 3
3
Slide 10- 7Copyright © 2011 Pearson Education, Inc.
Some roots, like are called irrational because we cannot express their exact value using rational numbers. In fact, writing with the radical sign is the only way we can express its exact value. However, we can approximate using rational numbers.
3
3
3
Approximating to two decimal places:
Approximating to three decimal places:
2 1.41
2 1.414
Note: Remember that the symbol, , means “approximately equal to.”
Slide 10- 8Copyright © 2011 Pearson Education, Inc.
Example 2
Approximate the roots using a calculator or table in the endpapers. Round to three decimal places.a.
b.
c.
18
Solution
32
Solution 32 5.657
18 4.243
3 56
Solution 3 56 3.826
Slide 10- 9Copyright © 2011 Pearson Education, Inc.
Example 3
Find the root. Assume variables represent nonnegative values.
b.
c.
d.
4y Solution
636m Solution
4y 2y
636m 36m
Because (y2)2 = y4.
Because (6m3)2 = 36m6.
10
4
36
25
x
ySolution
10 5
4 2
36 6
25 5
x x
y y
Slide 10- 10Copyright © 2011 Pearson Education, Inc.
continued
Find the root. Assume variables represent nonnegative values.
e.
f.
93 y Solution
164 81x Solution
93 y
164 81x 43x
3y
Slide 10- 11Copyright © 2011 Pearson Education, Inc.
Example 4
Find the root. Assume variables represent any real number.
a.
b.
c.
14y Solution
1036y Solution
14y 7y
56 y1036y
2( 3)n Solution 3n 2( 3)n
Slide 10- 12Copyright © 2011 Pearson Education, Inc.
continued
Find the root. Assume variables represent any real number.
d.
e.
c.
1249y Solution
3 927n Solution
1249y 67 y
33n
33 ( 4)w Solution
3 927n
33 ( 4)w 4w
Slide 10- 13Copyright © 2011 Pearson Education, Inc.
Radical function: A function containing a radical expression whose radicand has a variable.
Example 5a
Solution
Given f(x) = find f(3). 5 8,x
To find f(3), substitute 3 for x and simplify.
3 5 3 8f 15 8 7
Slide 10- 14Copyright © 2011 Pearson Education, Inc.
Example 6Find the domain of each of the following. a.
b.
8f x x
Solution Since the index is even, the radicand must be nonnegative.
Solution The radicand must be nonnegative.
8 0x 8x
3 9f x x
3 9 0x 3 9x
3x
Domain: 8 , or [8, )x x
Domain: 3 , or ( ,3]x x
Conclusion The domain of a radical function with an even index must contain values that keep its radicand nonnegative.
Slide 10- 15Copyright © 2011 Pearson Education, Inc.
Example 7
If you drop an object, the time (t) it takes in seconds to fall d feet is given by . Find the time ittakes for an object to fall 800 feet.
16dt
Understand We are to find the time it takes for an object to fall 800 feet.
Plan Use the formula , replacing d with 800. 16dt
Execute 80016t
50t
7.071t
Replace d with 800.
Divide within the radical.
Evaluate the square root.
Slide 10- 16Copyright © 2011 Pearson Education, Inc.
continued
Answer It takes an object 7.071 seconds to fall 800 feet.
Check We can verify the calculations, which we will leave to the viewer.
Slide 10- 17Copyright © 2011 Pearson Education, Inc.
For which square root is –12.37 the approximation for?
a)
b)
c)
d)
3.517
3.517
153
153
Slide 10- 18Copyright © 2011 Pearson Education, Inc.
For which square root is –12.37 the approximation for?
a)
b)
c)
d)
3.517
3.517
153
153
Slide 10- 19Copyright © 2011 Pearson Education, Inc.
Evaluate.
a) 0.2
b) 0.02
c) 0.002
d) 0.0002
0.0004
Slide 10- 20Copyright © 2011 Pearson Education, Inc.
Evaluate.
a) 0.2
b) 0.02
c) 0.002
d) 0.0002
0.0004
Slide 10- 21Copyright © 2011 Pearson Education, Inc.
Find the domain of f(x) = .
a)
b)
c)
d)
4 16x
4 , or ( , 4]x x
4 , or [4, )x x
4 , or [ 4, )x x
4 , or ( , 4]x x
Slide 10- 22Copyright © 2011 Pearson Education, Inc.
Find the domain of f(x) = .
a)
b)
c)
d)
4 16x
4 , or ( , 4]x x
4 , or [4, )x x
4 , or [ 4, )x x
4 , or ( , 4]x x
Copyright © 2011 Pearson Education, Inc.
Rational Exponents10.210.2
1. Evaluate rational exponents.2. Write radicals as expressions raised to rational
exponents.3. Simplify expressions with rational number exponents
using the rules of exponents.4. Use rational exponents to simplify radical expressions.
Slide 10- 24Copyright © 2011 Pearson Education, Inc.
Rational exponent: An exponent that is a rational number.
Rational Exponents with a Numerator of 1
a1/n = where n is a natural number other than 1.,n a
Note: If a is negative and n is odd, then the root is negative.If a is negative and n is even, then there is no real number root.
Slide 10- 25Copyright © 2011 Pearson Education, Inc.
Example 1
Rewrite using radicals, then simplify if possible. a. 491/2 b. 6251/4 c. (216)1/3
Solution
a.
b.
c.
1/ 249
1/4625
1/3216
49 7
4 625 5
3 216 6
Slide 10- 26Copyright © 2011 Pearson Education, Inc.
continued
Rewrite using radicals, then simplify. d. (16)1/4 e. 491/2 f. y1/6
Solution
d.
e.
f.
1/4( 16)
1/249
1/6y
4 16 There is no real number answer.
49 7
6 y
Slide 10- 27Copyright © 2011 Pearson Education, Inc.
continued
Rewrite using radicals, then simplify. g. (100x8)1/2 h. 9y1/5 i.
Solution
d.
e.
f.
8 1/2(100 )x
1/59y1/28
49
w
8 4100 10x x
59 y
8 4
49 7
w w
1/28
49
w
Slide 10- 28Copyright © 2011 Pearson Education, Inc.
General Rule for Rational Exponents
where a 0 and m and n are natural numbers other than 1.
/ ,m
nm n m na a a
Slide 10- 29Copyright © 2011 Pearson Education, Inc.
Example 2
Rewrite using radicals, then simplify, if possible. a. 272/3 b. 2433/5 c. 95/2
Solutiona.
b.
c.
2/3 1/3 227 (27 )
3/5 1/5 3243 (243 )
5/2 1/2 59 (9 )
23( 27) 23 9
35( 243) 33 27
5(3) 243 5( 9)
Slide 10- 30Copyright © 2011 Pearson Education, Inc.
continued
Rewrite using radicals, then simplify, if possible. d. e. f.
Solutiond.
e.
f.
33/21 1
16 16
52/5 2x x
3/5 35(4 1) (4 1)x x
31
4
1
64
3/21
16
2/5x 3/5(4 1)x
Slide 10- 31Copyright © 2011 Pearson Education, Inc.
Negative Rational Exponents
where a 0, and m and n are natural numbers with n 1.
//
1,m n
m na
a
Slide 10- 32Copyright © 2011 Pearson Education, Inc.
Example 3
Rewrite using radicals; then simplify if possible. a. 251/2 b. 272/3
Solutiona.
b.
1/ 21/ 2
125
25
23
1
27
1 1
525
2/32/3
127
27 2
1 1
3 9
Slide 10- 33Copyright © 2011 Pearson Education, Inc.
continued
Rewrite using radicals; then simplify if possible. c. d.
Solutionc.
1/2
1/2
25 1
36 2536
2/3
1
( 27)
1
2536
2/3d. ( 27)
23
1
( 27)
1/225
36
156
6
5
2/3( 27)
2
1
( 3)
1
9
Slide 10- 34Copyright © 2011 Pearson Education, Inc.
Example 4
Write each of the following in exponential form.
a.
Solution
6 5x
6 5x 5/ 6x
b. 34
1
x
a.
b. 34
1
x
3/4
1
x3/4x
Slide 10- 35Copyright © 2011 Pearson Education, Inc.
continued
Write each of the following in exponential form.
c.
Solution
45 x
45 x 4/5x
d. 34 5 2x
c.
d. 34 5 2x 3/ 45 2x
Slide 10- 36Copyright © 2011 Pearson Education, Inc.
Rules of Exponents Summary(Assume that no denominators are 0, that a and b are real
numbers, and that m and n are integers.)Zero as an exponent: a0 = 1, where a 0.
00 is indeterminate.Negative exponents:
Product rule for exponents:Quotient rule for exponents:Raising a power to a power:Raising a product to a power:Raising a quotient to a power:
1 , n
n
aa
m n m na a a
m n m na a a
nm mna a
n n nab a b
n na bb a
n
n
na ab b
1 ,n
n
aa
Slide 10- 37Copyright © 2011 Pearson Education, Inc.
Example 5a
Use the rules of exponents to simplify. Write the answer with positive exponents.
Solution
3/ 4 1/ 4y y
3/ 4 1/ 4y y 3/ 4 ( 1/ 4)y 2/ 4y1/ 2y
Use the product rule for exponents. (Add the exponents.)
Add the exponents.
Simplify the rational exponent.
Slide 10- 38Copyright © 2011 Pearson Education, Inc.
Example 5b
Use the rules of exponents to simplify. Write the answer with positive exponents.
Solution
1/3 1/63 4a a
1/3 1/63 4a a 1/3 1/612a 2/6 1/612a
3/6 1/212 or 12a a
Use the product rule for exponents. (Add the exponents.)
Rewrite the exponents with a common denominator of 6.
Add the exponents.
Slide 10- 39Copyright © 2011 Pearson Education, Inc.
Example 5c
Use the rules of exponents to simplify. Write the answer with positive exponents.
Solution
Use the quotient for exponents. (Subtract the exponents.)
Rewrite the subtraction as addition.
Add the exponents.
5/ 6
1/ 6
y
y
5/ 6
1/ 6
y
y 5/ 6 ( 1/ 6)y
5/ 6 1/ 6y
y
Slide 10- 40Copyright © 2011 Pearson Education, Inc.
Example 5d
Use the rules of exponents to simplify. Write the answer with positive exponents.
Solution
Add the exponents.
2/5 3/53 5y y
2/5 3/53 5y y 2/5 3/515y
1/515y
Slide 10- 41Copyright © 2011 Pearson Education, Inc.
Example 5e
Use the rules of exponents to simplify. Write the answer with positive exponents.
Solution
27/8m
27/8m (7/8)2m7/4m
Slide 10- 42Copyright © 2011 Pearson Education, Inc.
Example 5f
Use the rules of exponents to simplify. Write the answer with positive exponents.
Solution
32/5 4/53a b
32/5 4/53a b 3 2/5 3 4/5 33 ( ) ( )a b
(2/5)3 (4/5)327a b
6/5 12/527a b
Slide 10- 43Copyright © 2011 Pearson Education, Inc.
Example 5g
Use the rules of exponents to simplify. Write the answer with positive exponents.
Solution
8/3 3
6
(2 )x
x
8/3 3
6
(2 )x
x
3 8/3 3
6
2 ( )x
x
8
6
8x
x
8 68x 28x
Slide 10- 44Copyright © 2011 Pearson Education, Inc.
Example 6
Rewrite as a radical with a smaller root index. Assume that all variables represent nonnegative values.
a. b.
Solution
4 64
4a. 64 1/4642 1/4(8 )
21/48 1/28
6 10x
8
6 10b. x 10/6x5/3x
3 5x
3 2x x
Slide 10- 45Copyright © 2011 Pearson Education, Inc.
continued
Rewrite as a radical with a smaller root index. Assume that all variables represent nonnegative values.
c.
Solution
6 28 w y
6 28c. w y6 2 1/8( )w y
61/8 21/8w y
3/4 1/4w y
1/43w y
34 w y
Slide 10- 46Copyright © 2011 Pearson Education, Inc.
Example 7
Perform the indicated operations. Write the result using a radical.
Solution
b.a. 34x x
a. 34x x 1/ 2 3/ 4x x 1/ 2 3/ 4x 2/ 4 3/ 4x 5/ 4x
54 x
6 7
3
x
x
b.6 7
3
x
x
7 / 6
1/3
x
x
7 / 6 1/3x 7 / 6 2/ 6x
5/ 6x6 5x
Slide 10- 47Copyright © 2011 Pearson Education, Inc.
continued
Perform the indicated operations. Write the result using a radical.
Solution
c. 45 4c. 45 4 1/2 1/45 4
2/4 1/45 4
1/425 4 1/4(25 4)
1/41004 100
Slide 10- 48Copyright © 2011 Pearson Education, Inc.
Example 8
Write the expression below as a single radical. Assume that all variables represent nonnegative values.
Solution
4 x
4 x 1/2 1/4( )x(1/2)(1/4)x1/8x
8 x
Slide 10- 49Copyright © 2011 Pearson Education, Inc.
Simplify.
a)
b)
c)
d)
3/ 41/ 2 2 /3x y
3/8 1/ 4x y
3/8 3/ 4x y
3/8 1/ 2x y
3/ 4 1/ 2x y
Slide 10- 50Copyright © 2011 Pearson Education, Inc.
Simplify.
a)
b)
c)
d)
3/ 41/ 2 2 /3x y
3/8 1/ 4x y
3/8 3/ 4x y
3/8 1/ 2x y
3/ 4 1/ 2x y
Slide 10- 51Copyright © 2011 Pearson Education, Inc.
Simplify.
a) 5
b) 25
c) 25
d) 5
3 325 5
Slide 10- 52Copyright © 2011 Pearson Education, Inc.
Simplify.
a) 5
b) 25
c) 25
d) 5
3 325 5
Slide 10- 53Copyright © 2011 Pearson Education, Inc.
Simplify.
a) 4
b)
c) 4
d)
2/38
1
4
1
4
Slide 10- 54Copyright © 2011 Pearson Education, Inc.
Simplify.
a) 4
b)
c) 4
d)
2/38
1
4
1
4
Copyright © 2011 Pearson Education, Inc.
Multiplying, Dividing, and Simplifying Radicals10.310.3
1. Multiply radical expressions.2. Divide radical expressions.3. Use the product rule to simplify radical expressions.
Slide 10- 56Copyright © 2011 Pearson Education, Inc.
Product Rule for Radicals
If both and are real numbers, then
.n n na b a b
n a n b
Slide 10- 57Copyright © 2011 Pearson Education, Inc.
Example 1
Find the product and simplify. Assume all variables represent positive values. a. b.
Solutiona. b.
4 9 7 y
4 9 4 9
36
6
7 y 7y
Slide 10- 58Copyright © 2011 Pearson Education, Inc.
continued
Find the product and simplify. Assume all variables represent positive values. c. d.
Solution
c. d.
4 42 8 3 34 5x x
4 42 8 4 2 8
4 16
2
3 34 5x x 3 4 5x x
3 220x
Slide 10- 59Copyright © 2011 Pearson Education, Inc.
continued
Find the product and simplify. Assume all variables represent positive values. e. f.
Solution
e. f.
6
5
y
w
6
5
y
w
6
5
y
w
6
5
y
w
5 97 7y y
5 97 7y y 5 97 y y
147 y
2y
Slide 10- 60Copyright © 2011 Pearson Education, Inc.
continued
Find the product and simplify. Assume all variables represent positive values. g.
Solution
g.
2 2x x
2 2x x 2 2x x
4x
2x
Slide 10- 61Copyright © 2011 Pearson Education, Inc.
Raising an nth Root to the nth PowerFor any nonnegative real number a,
, where 0.n
nn
a ab
bb
Quotient Rule for RadicalsIf both and are real numbers, then
.n
n a a
n a n b
Slide 10- 62Copyright © 2011 Pearson Education, Inc.
Example 2Simplify. Assume variables represent positive values.a.
Solution
147
3
147
349 7
147
3
b.
b.
11
49
a. 11
49
11
49 11
7
c. 36
15
x
c. 36
15
x
3
3 6
15
x
3
2
15
x
Slide 10- 63Copyright © 2011 Pearson Education, Inc.
continuedSimplify. Assume variables represent positive values.d.
Solution
5
1024
x
5
1024
x 5
4
x
5
5 1024
x
e.
e.
5
5
12
4
d.5
5
12
45
12
4 5 3
Slide 10- 64Copyright © 2011 Pearson Education, Inc.
Simplifying nth Roots To simplify an nth root, 1. Write the radicand as a product of the greatest
possible perfect nth power and a number or an expression that has no perfect nth power factors.
2. Use the product rule when a is the perfect nth power.
3. Find the nth root of the perfect nth power radicand.
n n nab a b
Slide 10- 65Copyright © 2011 Pearson Education, Inc.
Example 3
Simplify. a. b.
Solution
80
80 16 5
16 5
4 5
6 98
6 49 2
6 49 2
6 7 2
Solution
6 98
42 2
Slide 10- 66Copyright © 2011 Pearson Education, Inc.
continued
Simplify. c. d.
Solution
34 448
3 34 64 7
34 4 7
316 7
Solution34 448
45 48
45 48 45 16 3
4 45 16 3
45 2 3
410 3
Slide 10- 67Copyright © 2011 Pearson Education, Inc.
Example 4a
Simplify the radical using prime factorization.
Solution
7 7 7 2
7 7 2
7 14
686Write 686 as a product of its prime factors.
The square root of the pair of 7s is 7.
Multiply the prime factors in the radicand.
Slide 10- 68Copyright © 2011 Pearson Education, Inc.
continued
Simplify the radical using prime factorization. b. c.
b.
Solution
3 2 2 5 5 5
35 2 2
35 4
3 500
3 500 4 810
c. 4 810 4 2 3 3 3 3 5
43 2 5
43 10
Slide 10- 69Copyright © 2011 Pearson Education, Inc.
Example 5a
Simplify.
Solution
532x
416 2 x x
416 2x x
24 2 x x
532x 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.
Slide 10- 70Copyright © 2011 Pearson Education, Inc.
Example 5b
Simplify
Solution
42 96 .a b
42 16 6 a b
42 16 6a b
22 4 6 a b
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.
42 96a b
28 6 a b Multiply 2 and 4.
Slide 10- 71Copyright © 2011 Pearson Education, Inc.
continued
Simplify. c. d.
Solution
3 103y y
3 103y y 3 93 y y y
3 93 3 y y y
3 3 3y y y
11 145 486x y
10 10 45 243 2 x x y y
Solution
11 145 486x y
6 3y y
10 10 45 5243 2x y x y
2 2 453 2x y xy
Slide 10- 72Copyright © 2011 Pearson Education, Inc.
Example 6
Find the product or quotient and simplify the results. Assume that variables represent positive values.a. b.
Solution
5 8 5 44 5 5 50x x
Solution5 8 40
4 10
2 10
5 44 5 5 50x x5 44 5 5 50x x
920 250x820 25 10 x x
420 5 10x x 4100 10x x
Slide 10- 73Copyright © 2011 Pearson Education, Inc.
continuedFind the product or quotient and simplify the results. Assume that variables represent positive values.c. d.
Solution
300
4
9 6
5
9 245
3 5
a b
a bSolution
300
4
300
4
75
25 3
9 6
5
9 245
3 5
a b
a b
9 6
5
2453
5
a b
a b
4 53 49a b4 43 49a b b
2 23 7a b b 2 221a b b5 3
Slide 10- 74Copyright © 2011 Pearson Education, Inc.
Simplify. Assume all variables represent nonnegative numbers.
a)
b)
c)
d)
2 33 9m m
3 3m m
23 3m m
627m
2 33 3m m
Slide 10- 75Copyright © 2011 Pearson Education, Inc.
Simplify. Assume all variables represent nonnegative numbers.
a)
b)
c)
d)
2 33 9m m
3 3m m
23 3m m
627m
2 33 3m m
Slide 10- 76Copyright © 2011 Pearson Education, Inc.
Simplify.
a)
b)
c)
d)
486
6 9
3 54
9 6
18 27
Slide 10- 77Copyright © 2011 Pearson Education, Inc.
Simplify.
a)
b)
c)
d)
486
6 9
3 54
9 6
18 27
Copyright © 2011 Pearson Education, Inc.
Adding, Subtracting, and Multiplying Radical Expressions10.410.4
1. Add or subtract like radicals.2. Use the distributive property in expressions containing
radicals.3. Simplify radical expressions that contain mixed
operations.
Slide 10- 79Copyright © 2011 Pearson Education, Inc.
Like radicals: Radical expressions with identical radicands and identical root indices.
Adding Like Radicals
To add or subtract like radicals, add or subtract the coefficients and leave the radical parts the same.
Slide 10- 80Copyright © 2011 Pearson Education, Inc.
Example 1
Add or subtract. a. b.
6 7 3 7
Solution
3 34 5 2 5
a.
b.
6 7 3 7 (6 3) 7
3 34 5 2 5 3(4 2) 5
36 5
9 7
Slide 10- 81Copyright © 2011 Pearson Education, Inc.
continued
Simplify.c. d.
3 37 5 12 5x x
Solution
14 5 11 2 11 5 5
c.
d.
3 37 5 12 5x x 35 5x Combine the like radicals by subtracting the coefficients and keeping the radical.
14 5 11 2 11 5 5
(14 5) 5 (2 1) 11
9 5 11
Regroup the terms.
Slide 10- 82Copyright © 2011 Pearson Education, Inc.
Example 2
Add or subtract. a. b.
28 7
Solution
3 34 135 2 5
a.
b.
4 7 7 Factor 28.28 7
2 7 7 3 7 Combine like radicals.
Simplify.
3 34 135 2 5 3 34 27 5 2 5 3 34 3 5 2 5
3 312 5 2 5 310 5
Slide 10- 83Copyright © 2011 Pearson Education, Inc.
continued
c.
Solution
5 5 563 112 28x x x
c.5 5 563 112 28x x x 4 4 49 7 16 7 4 7x x x x x x
2 2 23 7 4 7 2 7x x x x x x 23 7x x
Slide 10- 84Copyright © 2011 Pearson Education, Inc.
Example 3a
Find the product.
Solution
3 6 5 7 7
3 6 5 7 7 3 6 5 3 6 7 7
3 30 21 42
Use the distributive property.
Multiply.
Slide 10- 85Copyright © 2011 Pearson Education, Inc.
Example 3c
Find the product.
Solution
Use the product rule.
4 5 2 5 5 2 .
4 5 2 5 5 2
4 5 5 5 4 5 2 5 2 5 2 2
4 5 20 10 10 5 2
20 20 10 10 10
10 19 10
Use the distributive property.
Find the products.
Combine like radicals.
Slide 10- 86Copyright © 2011 Pearson Education, Inc.
Example 3d
Find the product.
Solution
4 3 7x y x y
4 3 7x y x y
4 3 4 7 3 7x x x y y x y y
12 28 3 7x xy xy y
12 25 7x xy y
Slide 10- 87Copyright © 2011 Pearson Education, Inc.
Example 3e
Find the product.
Solution
2
5 3
2 2
5 2 5 3 3
5 2 15 3
8 2 15
Simplify.
2
5 3 Use (a – b)2 = a2 – 2ab – b2.
Slide 10- 88Copyright © 2011 Pearson Education, Inc.
Example 4a
Find the product.
Solution
8 3 8 3
228 3
61
8 3 8 3
64 3 Simplify.
Use (a + b)(a – b) = a2 – b2.
Slide 10- 89Copyright © 2011 Pearson Education, Inc.
Example 4b
Find the product.
Solution
7 2 3 7 2 3
2 2
7 2 3
7 12
7 2 3 7 2 3
7 4 3
5
Slide 10- 90Copyright © 2011 Pearson Education, Inc.
Example 5
Simplify.a.
Solution
9650
3
a. b.
2 14 3 21
9650
3 32 25 2
16 2 5 2
16 2 5 2
4 2 5 2
9 2
2 14 3 21
2 14 3 21 28 63
4 7 9 7 2 7 3 7
5 7
b.
Slide 10- 91Copyright © 2011 Pearson Education, Inc.
Simplify.
a)
b)
c)
d)
2 12 5 27 48 2 3
9 3
17 3
71 3
25 3
Slide 10- 92Copyright © 2011 Pearson Education, Inc.
Simplify.
a)
b)
c)
d)
2 12 5 27 48 2 3
9 3
17 3
71 3
25 3
Slide 10- 93Copyright © 2011 Pearson Education, Inc.
Multiply.
a)
b)
c)
d)
3 2 3 2 5 3
9 15 6 3 2
6 3 2 15 3 15 6
5 14 6 15 3
25 3
Slide 10- 94Copyright © 2011 Pearson Education, Inc.
Multiply.
a)
b)
c)
d)
3 2 3 2 5 3
9 15 6 3 2
5 14 6 15 3
25 3
6 3 2 15 3 15 6
Copyright © 2011 Pearson Education, Inc.
Rationalizing Numerators and Denominators of Radical Expressions10.510.5
1. Rationalize denominators.2. Rationalize denominators that have a sum or difference
with a square root term.3. Rationalize numerators.
Slide 10- 96Copyright © 2011 Pearson Education, Inc.
Example 1a
Rationalize the denominator.
Solution
Simplify.
8
5
Multiply by
8 5
25
8
5
8 5
5 5 5
.5
8 5
5
Slide 10- 97Copyright © 2011 Pearson Education, Inc.
Example 1b
Rationalize the denominator.
SolutionUse the quotient rule for square roots to separate the numerator and denominator into two radicals.
2
32
3
2
3
2 3
3 3
6
9
6
3
Multiply by 3
.3
Simplify.
Warning: Never divide out factors common to a radicand and a number not under a radical.
Slide 10- 98Copyright © 2011 Pearson Education, Inc.
Example 1c
Rationalize the denominator.
Solution
5
3x
3
5
3
3
xx
x
5
3x
2
5 3
9
x
x
5 3
3
x
x
Slide 10- 99Copyright © 2011 Pearson Education, Inc.
Rationalizing DenominatorsTo rationalize a denominator containing a single nth root, multiply the fraction by a well chosen 1 so that the product’s denominator has a radicand that is a perfect nth power.
Slide 10- 100Copyright © 2011 Pearson Education, Inc.
Example 2a
Rationalize the denominator. Assume that variables represent positive values.
Solution
3
5
3
3
5
3 33
35
3
9
9
3
3
5 9
27
35 9
3
Slide 10- 101Copyright © 2011 Pearson Education, Inc.
Example 2b
Rationalize the denominator. Assume that variables represent positive values.
Solution
3
3
w
z
3
3
w
z
2
3
3
23
3 zw
z z
3 2
3 3
wz
z
3 2wz
z
Slide 10- 102Copyright © 2011 Pearson Education, Inc.
Example 2c
Rationalize the denominator. Assume that variables represent positive values.
Solution
32
7
16x
32
7
16x
3
3 2
7
16x
3
3 2
3
3
7
1 46
4
x
x
x
3
3 3
28
64
x
x
3 28
4
x
x
Slide 10- 103Copyright © 2011 Pearson Education, Inc.
Rationalizing a Denominator Containing a Sum or DifferenceTo 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.
Slide 10- 104Copyright © 2011 Pearson Education, Inc.
Example 3a
Rationalize the denominator and simplify. Assume variables represent positive values.
Solution
7
3 57
3 53 5
3 55 3
7
2 2
7( 3 5)
( 3) (5)
7 3 35
3 25
7 3 35
22
1(7 3 35)
1(22)
35 7 3
22
Slide 10- 105Copyright © 2011 Pearson Education, Inc.
Example 3b
Rationalize the denominator and simplify. Assume variables represent positive values.
Solution
12 5
11 312 5
11 3
11 3
11 3
12 5
11 3
2 2
12 5 11 3
( 11) ( 3)
12 55 12 15
11 3
12 55 12 15
8
4(3 55 3 15)
8
3 55 3 15
2
Slide 10- 106Copyright © 2011 Pearson Education, Inc.
Example 3c
Rationalize the denominator and simplify. Assume variables represent positive values.
Solution
4
3x
4
3x 4 3
3 3
x
x x
2 2
4 3
3
x
x
4 12
9
x
x
Slide 10- 107Copyright © 2011 Pearson Education, Inc.
Example 4a
Rationalize the numerator. Assume variables represent positive values.
Solution
3
8
x
3
8
x
3
3 3
8 x
x x
29
8 3
x
x
3
8 3
x
x
Slide 10- 108Copyright © 2011 Pearson Education, Inc.
Example 4b
Rationalize the numerator. Assume variables represent positive values.
Solution
5 3
6
x
5 3
6
x 5 3 5 3
6 5 3
x x
x
225 3
6 5 3
x
x
25 3
30 6 3
x
x
Slide 10- 109Copyright © 2011 Pearson Education, Inc.
Rationalize the denominator.
a)
b)
c)
d)
8
3
2 3
33 2
3
2 6
3
3 6
3
Slide 10- 110Copyright © 2011 Pearson Education, Inc.
Rationalize the denominator.
a)
b)
c)
d)
8
3
2 3
33 2
3
2 6
3
3 6
3
Slide 10- 111Copyright © 2011 Pearson Education, Inc.
Rationalize the denominator.
a)
b)
c)
d)
5
13 7
5 13 5 7
6
5 13 5 7
6
13 7
5
13 7
5
Slide 10- 112Copyright © 2011 Pearson Education, Inc.
Rationalize the denominator.
a)
b)
c)
d)
5
13 7
5 13 5 7
6
5 13 5 7
6
13 7
5
13 7
5
Copyright © 2011 Pearson Education, Inc.
Radical Equations and Problem Solving10.610.6
1. Use the power rule to solve radical equations.
Slide 10- 114Copyright © 2011 Pearson Education, Inc.
Power Rule for Solving EquationsIf both sides of an equation are raised to the same integer power, the resulting equation contains all solutions of the original equation and perhaps some solutions that do not solve the original equation. That is, the solutions of the equation a = b are contained among the solutions of an = bn, where n is an integer.
Radical equation: An equation containing at least one radical expression whose radicand has a variable.
Slide 10- 115Copyright © 2011 Pearson Education, Inc.
Example 1
Solve. a. b.
Solution a.
12y 3 4x
2212y
144y
144 12
12 12
Check
True
b. 3 33 4x
64x
Check3 64 4
4 4 True
Slide 10- 116Copyright © 2011 Pearson Education, Inc.
Solution
Example 2a Solve.5 6x
5 6x
225 (6)x
5 36x
41x
Check:
The number 41 checks. The solution is 41.
5 6x
41 5 6
36 6
6 6
Slide 10- 117Copyright © 2011 Pearson Education, Inc.
Solution
Example 2b Solve.3 4 2x
3 4 2x
333 4 ( 2)x
4 8x
4x
Check:
True. The solution is 4.
3 4 2x
3 4 4 2
3 8 2
2 2
Slide 10- 118Copyright © 2011 Pearson Education, Inc.
Solution
Example 2c Solve.4 1 5x
4 1 5x
224 1 ( 5)x
4 1 25x
4 24x
Check:
False, so 6 is extraneous. This equation has no real number solution.
4 1 5x
4(6) 1 5
25 5
5 5
6x
Slide 10- 119Copyright © 2011 Pearson Education, Inc.
Solution
Example 3a Solve.4 60x x
4 60x x
2 2
4 60x x
224 60x x
16 60x x
15 60x
4x
Check:
The number 4 checks. The solution is 4.
4 60x x
4 4 4 60
4 2 64
8 8
Slide 10- 120Copyright © 2011 Pearson Education, Inc.
Example 4Solve.Solution
5 7x x
5 7x x
225 7x x
2 10 25 7x x x 2 11 25 7x x 2 11 18 0x x
( 2)( 9) 0x x 2 0 or 9 0x x
2x 9x
Square both sides.
Use FOIL.
Subtract x from both sides.
Factor.
Use the zero-factor theorem.
Subtract 7 from both sides.
Slide 10- 121Copyright © 2011 Pearson Education, Inc.
continued
Checks 2x 9x
5 7x x
2 5 2 7
3 9
3 3 False.
9 5 9 7
4 16
4 4 True.
Because 2 does not check, it is an extraneous solution. The only solution is 9.
Slide 10- 122Copyright © 2011 Pearson Education, Inc.
Example 5a
Solve.
SolutionCheck
This solution does not check, so it is an extraneous solution. The equation has no real number solution.
4 6x
4 6x
2x
2 2
2x
4x
4 6x
4 4 6
2 4 6 2 6
Slide 10- 123Copyright © 2011 Pearson Education, Inc.
Example 5b
Solve 4 3 3 5.x
Solution
Check
The solution set is 13.
4 3 3 5x 4 3 2x
444 3 2x
3 16x
13x
4 3 3 5x
4 13 3 3 5
4 16 3 5
2 3 5
5 5
Slide 10- 124Copyright © 2011 Pearson Education, Inc.
Example 6
Solve 16 8x x
SolutionCheck
There is no solution.
16 8x x
2 2
16 8x x
16 8 8 64x x x x
16 16 64x x x
48 16 x
16 8x x
9 16 9 8
25 3 8
5 11
3 x 2 2( 3) ( )x 9 x
Slide 10- 125Copyright © 2011 Pearson Education, Inc.
Solving Radical EquationsTo solve a radical equation,1. Isolate the radical if necessary. (If there is more
than one radical term, isolate one of the radical terms.)
2. Raise both sides of the equation to the same power as the root index of the isolated radical.
3. If all radicals have been eliminated, solve. If a radical term remains, 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.
Slide 10- 126Copyright © 2011 Pearson Education, Inc.
Solve.
d) no solution
a) 6
b) 8
c) 9
5 4 2x x
Slide 10- 127Copyright © 2011 Pearson Education, Inc.
Solve.
d) no solution
a) 6
b) 8
c) 9
5 4 2x x
Slide 10- 128Copyright © 2011 Pearson Education, Inc.
Solve.
d) no real-number solution
a) 2
b) 4
c)
4 10 2a a
12
5
Slide 10- 129Copyright © 2011 Pearson Education, Inc.
Solve.
d) no real-number solution
a) 2
b) 4
c)
4 10 2a a
12
5
Slide 10- 130Copyright © 2011 Pearson Education, Inc.
Solve.
d) no real-number solution
a) 3, 4
b) 3
c) 4
2 5 16x x
Slide 10- 131Copyright © 2011 Pearson Education, Inc.
Solve.
d) no real-number solution
a) 3, 4
b) 3
c) 4
2 5 16x x
Copyright © 2011 Pearson Education, Inc.
Complex Numbers10.710.7
1. Write imaginary numbers using i.2. Perform arithmetic operations with complex numbers.3. Raise i to powers.
Slide 10- 133Copyright © 2011 Pearson Education, Inc.
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.
1i
Slide 10- 134Copyright © 2011 Pearson Education, Inc.
Example 1
Write each imaginary number as a product of a real number and i.a. b. c.
Solutiona. b. c.
16 21 32
16 21 32
1 16
1 16 4i
4i
1 21
1 21 21i
1 32
1 32
16 2i 4 2i
Slide 10- 135Copyright © 2011 Pearson Education, Inc.
Rewriting Imaginary NumbersTo write an imaginary number in terms of the imaginary unit i,1. Separate the radical into two factors, 2. Replace with i.3. Simplify
n
1 .n 1.n
Slide 10- 136Copyright © 2011 Pearson Education, Inc.
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.
Slide 10- 137Copyright © 2011 Pearson Education, Inc.
Example 2a
Add or subtract. (9 + 6i) + (6 – 13i)
SolutionWe add complex numbers just like we add polynomials—by combining like terms.
(9 + 6i) + (6 – 13i) = (9 + 6) + (6i – 13i ) = –3 – 7i
Slide 10- 138Copyright © 2011 Pearson Education, Inc.
Example 2b
Add or subtract. (3 + 4i) – (4 – 12i)
SolutionWe subtract complex numbers just like we subtract polynomials.
(3 + 4i) – (4 – 12i) = (3 + 4i) + (4 + 12i) = 7 + 16i
Slide 10- 139Copyright © 2011 Pearson Education, Inc.
Example 3Multiply.a. (8i)(4i) b. (6i)(3 – 2i)
Solution a. (8i)(4i) b. (6i)(3 – 2i) 232i
)132(
32
218 12i i
18 1 ( 12 )i
18 12i
12 18i
Slide 10- 140Copyright © 2011 Pearson Education, Inc.
continuedMultiply.c. (9 – 4i)(3 + i) d. (7 – 2i)(7 + 2i)
Solution c. (9 – 4i)(3 + i) d. (7 – 2i)(7 + 2i)
227 9 12 4i i i
27 3 4 )1(i
27 3 4i
31 3i
249 14 14 4i ii
49 4 1( )
49 4
53
Slide 10- 141Copyright © 2011 Pearson Education, Inc.
Complex conjugate: The complex conjugate of a complex number a + bi is a – bi.
Slide 10- 142Copyright © 2011 Pearson Education, Inc.
Example 4a
Divide. Write in standard form.
Solution Rationalize the denominator.
7
3i
7
3i7
3
i
i i
2
7
3
i
i
7
3( 1)
i
7
3
i
7
3
i
Slide 10- 143Copyright © 2011 Pearson Education, Inc.
Example 4b
Divide. Write in standard form.
Solution Rationalize the denominator.
3 5
5
i
i
3 5
5
i
i
53
5
5
5
i
i
i
i
2
2
15 3 25 5
25
i i i
i
15 3 25 5( 1)
25 ( 1)
i i
15 3 25 5
25 1
i i
10 28
26
i
10 28
26 26
i
5 14
13 13
i
Slide 10- 144Copyright © 2011 Pearson Education, Inc.
Example 5
Simplify.
Solution
40 33a. b. i i
1040 4 10a. = = 1 i i = 1 Write i40 as (i4)10.
33 32b. = i i i
84 = i i
= 1 i
= i
Write i32 as (i4)8.
Replace i4 with 1.
Slide 10- 145Copyright © 2011 Pearson Education, Inc.
Simplify. (4 + 7i) – (2 + i)
a) 2 + 7i2
b) 2 + 8i
c) 6 + 6i
d) 6 + 8i
Slide 10- 146Copyright © 2011 Pearson Education, Inc.
Simplify. (4 + 7i) – (2 + i)
a) 2 + 7i2
b) 2 + 8i
c) 6 + 6i
d) 6 + 8i
Slide 10- 147Copyright © 2011 Pearson Education, Inc.
Multiply. (4 + 7i)(2 + i)
a) 15 + 10i
b) 1 + 10i
c) 15 + 18i
d) 15 + 18i
Slide 10- 148Copyright © 2011 Pearson Education, Inc.
Multiply. (4 + 7i)(2 + i)
a) 15 + 10i
b) 1 + 10i
c) 15 + 18i
d) 15 + 18i
Slide 10- 149Copyright © 2011 Pearson Education, Inc.
Write in standard form.
a)
b)
c)
d)
4
2 3
i
i
5 14
13 13
i
5 14
13 13
i
11 14
13 13
i
11 14
13 13
i
Slide 10- 150Copyright © 2011 Pearson Education, Inc.
Write in standard form.
a)
b)
c)
d)
4
2 3
i
i
5 14
13 13
i
5 14
13 13
i
11 14
13 13
i
11 14
13 13
i
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