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Math 102, Intermediate Algebra Name___________________________________ Author: Debra Griffin Circle one: Red Bluff Main Campus Burney Weaverville
Page 1 of 39
Chapter 4 Radical Expressions 4.1 Review of Powers and Roots 2
4.2 Simplifying Radicals 5 4.3 Operations with Radical Expressions 12 Multiplying/dividing, adding/subtracting radical expressions, rationalizing denominators
4.4 Rational Exponents 18 Convert between rational exponents and radical notation, simplify rational exponent expressions, simplify radical expressions
4.5 Equation Solving 26 Solve equations with radicals, applications of the Pythagorean Theorem
4.6 Complex Numbers 33 Simplifying complex numbers in radical form, add, subtract, multiply complex numbers, rationalize denominators of complex numbers, powers of i Lecture Note-Taking Guide Scoring Rubric Completion of these lecture notes for extra credit points is optional. Because it is an extra credit option, the expectation of work quality is very high. There is no partial extra credit for this assignment. Any of the following criteria that are not met will result in no extra credit. Extra Credit Points Condition
5 • complete • all worksteps shown • correct answers • neatly done in pencil • correctly ordered
• hole punched • fastened in folders with
fasteners • turned in on time • labeled with name and
site
0 Any of the required criteria have not been met
Math 102, Intermediate Algebra Section 4.1
Page 2 of 39
4.1 Review of Powers and Roots
Using Exponential Notation
Consider the expression 5 • 5 • 5
We can represent this product with
the following shorthand notation: The exponent, 3, indicates how many times the base, 5, is used as a factor.
We say 53 is written in exponential notation.
Demonstration Problems Practice Problems Write using exponential notation.
1. (a) 8 • 8 • 8 • 8 • 8
2. (a)
€
23
•23
•23• 23
Evaluate 3. (a) (–1.2)2
4. (a) – 12
⎛⎝⎜
⎞⎠⎟5
Write using exponential notation.
1. (b) 7 • 7 • 7 • 7
2. (b)
€
14
•14
•14
Evaluate 3. (b) (–1.1)2
4. (b) – 25
⎛⎝⎜
⎞⎠⎟3
Answers: 1. (b) 74; 2. (b) ( 14
)3; 3. (b) 1.21; 4. (b) – 8125
exponent
base
53 = 5 • 5 • 5
Math 102, Intermediate Algebra Section 4.1
Page 3 of 39
Square Roots a2 = a
Note that a is not a real number for a < 0. Problem Solution Answer Example (a) Calculate 9 9 = 32 = 3 9 = 3
Example (b) Calculate 49
49= 2
3⎛⎝⎜
⎞⎠⎟2
= 23
49= 23
Example (c) Calculate 3Round to 4 decimal places
In a calculator, press ⇒ 3⇒ = /enter 3 ≈ 1.7320508075688…
3 ≈ 1.7321
Example (d) Calculate −9 There is no real number such that its square is –9.
−9 is not a real number
Demonstration Problems Practice Problems
Calculate (round to 4 decimal places if necessary) 5. (a) 25
6. (a) 149
7. (a) 24
8. (a) − 4
Calculate (round to 4 decimal places if necessary) 5. (b) 36
6. (b) 9100
7. (b) 5 8. (b) − 9
Answers: 5. (b) 6; 6. (b) 310
; 7. (b) 2.2361; 8. (b) –3
Math 102, Intermediate Algebra Section 4.1
Page 4 of 39
nth Roots
Problem Solution Answer Example (a) Calculate 83 83 = 233 = 2 83 = 2
Example (b) Calculate 132
5 132
5 = 12
⎛⎝⎜
⎞⎠⎟5
5 = 12
132
5 = 12
Example (c) Calculate −273 −273 = (−3)33 = −3 −273 = −3
Example (d) Calculate 404 . Round to 4 decimal places
In a calculator, press 4 ⇒ ⇒ 40 ⇒ = /enter 404 ≈ 2.514866859…
404 ≈ 2.5149
Demonstration Problems Practice Problems
Calculate (round to 4 decimal places if necessary) 9. (a) 814
10. (a) − 8125
3
11. (a) 203
12. (a) 16
Calculate (round to 4 decimal places if necessary) 9. (b) 646
10. (b) − 18
3
11. (b) 503 12. (b) −15
Answers: 9. (b) 2; 10. (b) − 12
; 11. (b) 3.6840; 12. (b) –1
Index Radical Sign
Radicand
(when n is even, )
Math 102, Intermediate Algebra Section 4.2
Page 5 of 39
4.2 Simplifying Radicals
We say that a square root radical expression is in simplest form when
1. its radicand has no square factors 2. the radicand is not a fraction 3. no denominator contains a radical
Example (a) Which of the following radicals are in simplest form?
(i) 33 (ii) 20 (iii) 13
(iv) 15
33 = 3•11 The radicand has no square factors, so the radical
is in simplest form.
20 = 4 • 5 4 is a square factor of the radicand, so the radical
is not in simplest form.
The radicand is a fraction, so the radical
is not in simplest form.
The denominator contains a radical, so the expression
is not in simplest form.
To simplify square root radicals we can use the following rules of square roots:
Product Rule Quotient Rule
ab = a • b (for a ≥ 0 and b ≥ 0) ab= a
b (for a ≥ 0 and b > 0)
Example (b) Simplify 20 .
20 = 22 • 5
= 22 • 5
= 2 • 5
= 2 5
Example (c) Simplify 180 .
180 = 22 • 32 • 5
= 22 • 32 • 5
= 2 • 3• 5
= 6 5
Math 102, Intermediate Algebra Section 4.2
Page 6 of 39
Demonstration Problems Practice Problems Simplify
1. (a) 28 2. (a) 50 3. (a) 300
4. (a) 15049
Simplify
1. (b) 40 2. (b) 75 3. (b) 500
4. (b) 1759
Answers: 1. (b) 2 10 ; 2. (b) 5 3 ; 3. (b) 10 5 ; 4. (b) 5 73
Math 102, Intermediate Algebra Section 4.2
Page 7 of 39
In general, an nth root radical is considered to be in simplest form when
1. its radicand has no factors raised to a power greater than or equal to the index 2. the radicand is not a fraction 3. no denominator contains a radical
Example (d) Which of the following radicals are in simplest form?
(i) 183 (ii) 484 (iii) 14
5 (iv) 5 75
34
183 = 2 • 323 The radicand has no cube factors, so the radical
is in simplest form.
484 = 24 • 34 24 is 4th power factor in the radicand, so the radical
is not in simplest form.
The radicand is a fraction, so the radical
is not in simplest form.
The denominator contains a radical, so the radical
is not in simplest form.
To simplify nth root radicals we can use the following rules of nth roots:
Product Rule Quotient Rule
abn = an • bn (for a ≥ 0 and b ≥ 0 when n is even)
ab
n = an
bn
(for a ≥ 0 and b > 0 when n is even)
Example (e) Simplify 484 .
484 = 24 • 34
= 244 • 34
= 2 • 34
= 2 34
Example (f ) Simplify −645 .
−645 = −1• 265
= −1• 25 • 25
= −15 • 255 • 25
= −1• 2 • 25
= −2 25
Math 102, Intermediate Algebra Section 4.2
Page 8 of 39
Demonstration Problems Practice Problems
Simplify 5. (a) 403 6. (a) −813
7. (a) 1124
8. (a) − 965
Simplify 5. (b) 543 6. (b) −2503 7. (b) 1624 8. (b) − 1605
Answers: 5. (b) 3 23 ; 6. (b) −5 23 ; 7. (b) 3 24 ; 8. (b) −2 55
Math 102, Intermediate Algebra Section 4.2
Page 9 of 39
Variable Radical Expressions To simplify a radical that contains variables, we continue to use the rules: a2 = a (for a ≥ 0) ann = a (for a ≥ 0 when n is even)
abn = an • bn (for a ≥ 0 and b ≥ 0 when n is even)
ab
n = an
bn (for a ≥ 0 and b > 0 when n is even)
In all of the exercises in this chapter, assume all variables represent positive real numbers.
Example (g) Simplify 8y3 .
8y3 = 23y3
= 22 • 2 • y2 • y
= 22 • y2 • 2 • y
= 22 • y2 • 2 • y
= 2 • y • 2 • y
= 2y 2y
Example (h) Simplify x83 .
x83 = x3 • x3 • x23
= x33 • x33 • x23
= x • x • x23 = x2 x23
Alternative method:
x83 = x6 • x23
= x2( )3 • x23
= x2( )33 • x23
= x2 x23
Math 102, Intermediate Algebra Section 4.2
Page 10 of 39
Example (i) Simplify 9a2
49b2.
9a2
49b2= 9a2
49b2
= 32a2
72b2
= 32 • a2
72 • b2
= 3a7b
Demonstration Problems Practice Problems
Simplify 9. (a) 36a2b2
10. (a) 25x2
Simplify 9. (b) 4x4
10. (b) 8x3
3
Answers: 9. (b) 2x2; 10. (b) 2x ;
Math 102, Intermediate Algebra Section 4.2
Page 11 of 39
Demonstration Problems Practice Problems Simplify 11. (a) 27x53 12. (a) 40x5y6z8
13. (a) a5
16b84
Simplify 11. (b) 8a43
12. (b) 90x4y5z6
13. (b) a6
27b123
Answer: 11. (b) 2a a3 ; 12. (b) 3x2y2z3 10y ; 13. (b) a2
3b4
Math 102, Intermediate Algebra Section 4.3
Page 12 of 39
4.3 Operations with Radical Expressions
To simplify a radical expression that contains variables and any of the operations +, – , ×, or ÷, we continue to use the rules introduced in the previous section with one more:
Definition of square root a2 = a (for a ≥ 0)
Definition of nth root ann = a (for a ≥ 0 when n is even) Product Rule abn = an • bn (for a ≥ 0 and b ≥ 0 when n is even)
Quotient Rule
ab
n = an
bn (for a ≥ 0 when n is even and b ≠ 0
always)
Distributive Property a x + b x = (a + b) x In all of the exercises in this chapter, assume all variables represent positive real numbers.
Example (a) Simplify 7 • 7 .
7 • 7 = 49
= 72
= 7
Example (b) Simplify 9a4b23 • 24a5b83 .
9a4b23 • 24a5b83 = 9a4b2 • 24a5b83
= 216a9b103
= 23 • 33 • a9 •b9 •b3
= 233 • 333 • a93 • b93 • b3
= 2 • 3• a3 •b3 • b3
= 6a3b3 b3
Example (c) Simplify 4812
.
4812
= 4812
= 4
= 2
Math 102, Intermediate Algebra Section 4.3
Page 13 of 39
Demonstration Problems Practice Problems Simplify
1. (a) 50 • 2 2. (a) 2xy3 • 54xy
3. (a) 50010
4. (a) 75x2y48x4y9
Simplify
1. (b) 40 • 10 2. (b) 3x3y4 • 24xy
3. (b) 3507
4. (b) 72x6y8y
Answers: 1. (b) 20; 2. (b) 6x2y2 2y ; 3. (b) 5 2 ; 4. (b) 3x3
Math 102, Intermediate Algebra Section 4.3
Page 14 of 39
Example (d) Simplify 5 3 − 3 . 5 3 − 3 = 5 3 −1 3 = (5 −1) 3 = 4 3
Example (e) Simplify 5 x − x + x2
.
5 x − x + x2
= 5 x1
− x1
+ x2
= 5 x1• 22− x1• 22+ x2
= 10 x2
− 2 x2
+ 1 x2
= 9 x2
Example (f ) Simplify 8 12 − 4 27 .
8 12 − 4 27 = 8 4 • 3 − 4 9 • 3
= 8 • 2 3 − 4 • 3 3
= 16 3 −12 3
= 4 3
Example (g) Simplify x 50x + 2x3 .
x 50x + 2x3 = x 25 • 2x + 2x2 • x
= x • 5 2x + x 2x
= 5x 2x +1x 2x
= (5 +1)x 2x
= 6x 2x
Math 102, Intermediate Algebra Section 4.3
Page 15 of 39
Demonstration Problems Practice Problems
Simplify
5. (a) 5 + 52
− 54
6. (a) 2 40 + 90
7. (a) 3 x4y3 + 2x xy3 Multiply and simplify
8. (a) 3+ 2( ) 5 − 3 2( )
9. (a) 3 + 2 5( ) 3 − 2 5( )
Simplify
5. (b) 3 − 35
+ 325
6. (b) 2 45 + 6 20 7. (b) 5y xy3 − xy43 Multiply and simplify
8. (b) 6 + 3( ) 4 − 3( )
9. (b) 5 + 11( ) 5 − 11( )
Answers: 5. (b) 21 325
; 6. (b) 18 5 ; 7. (b) 4y xy3 ; 8. (b) 21− 2 3 ; 9. (b) –6
Math 102, Intermediate Algebra Section 4.3
Page 16 of 39
Rationalization Recall that a simplified radical expression cannot contain a radical in any denominator. There is a process to simplify such a radical called rationalization of the denominator. The process refers to manipulating an expression until the denominator contains only rational numbers. This process is used also to remove radical signs from variables in the denominator.
Example (h) Simplify 23
.
23= 2
3• 33
= 2 33 • 3
= 2 332
= 2 33
Recall that (a + b)(a – b) = a2 – b2 and a2 = a . These two properties allow us to simplify a radical denominator of the form a b + c d .
Notice that a b + c d( ) a b − c d( ) = a2b − ac bd + ac bd − c d • c d
= a2b − c2 d 2
= a2b − c2d
We call a b − c d the conjugate of a b + c d .
Example (i) Simplify 13+ 2
.
13+ 2
= 13+ 2
• 3− 23− 2
=1• 3− 2( )
3+ 2( ) 3− 2( )
= 3− 29 – 3• 2 + 3• 2 – 2 • 2
= 3− 29 – 2
= 3− 27
Math 102, Intermediate Algebra Section 4.3
Page 17 of 39
Demonstration Problems Practice Problems Simplify
10. (a) x2y
11. (a) 12 − 5
12. (a) x1+ 2 y
Simplify
10. (b) 53x
11. (b) 110 − 2
12. (b) a5 + 3 b
Answer: 10. (b) 5 3x3x
; 11. (b) 10 + 26 ; 12. (b)
5 a − 3 ab25 − 9b
Math 102, Intermediate Algebra Section 4.4
Page 18 of 39
4.4 Rational Exponents
Here is a review of exponent rules:
Rules for Exponents Examples Product Rule am • an = am+n 1. x12 • x3 =
Quotient Rule
€
am
an= am−n (a ≠ 0)
2.
€
x16
x 7 =
Power Rules
(am )n = amn (ab)m = am bm
€
ab"
# $ %
& ' m
=am
bm (b ≠ 0)
3. (x4)5 = 4. (2x)3 = 5.
€
2x"
# $ %
& ' 4
=
Zero Exponent a0 = 1 (a ≠ 0) 8.
€
35
35 = 9. (2xy)0 =
Negative Exponent a−n = 1
an (a ≠ 0)
10. x –5 =
Let’s explore using rule am • an = am+n. Fill in the boxes below.
From the inquiry above, it seems reasonable that since
312 = 3
we can then define
a12 = a (a ≥ 0)
33 • 33 = 36
3… . • 3… . = 34
3…. • 3…. = 32
3…. • 3…. = 31
Compare with:
• = 3
⇐ This uses the rule am • an = am+n. ⇐ Think: £ + £ = 4. ⇐ Think: £ + £ = 2. ⇐ Think: £ + £ = 1.
Math 102, Intermediate Algebra Section 4.4
Page 19 of 39
Let’s try another example
And now, it seems reasonable that since
513 = 53
that we can define
a13 = a3
In fact, we now have a new general rule to add to our rules of exponents list:
a1n = an (for a ≥ 0 when n is even)
Example (a) Write 11with fractional exponents.
11 = 1112
Example (b) Write x3 with fractional exponents.
x3 = x3( )12
= x3 •
12
= x32
From example (b), we can see that we can generalize our new rule as:
amn = amn (a ≥ 0 when n is even)
53 • 53 • 53 = 59
5… . • 5… • 5… = 56
5…. • 5… • 5… = 53
5…. • 5… • 5… = 51
Compare with:
• • = 5
⇐ Think: £ + £ + £ = 6. ⇐ Think: £ + £ + £ = 3. ⇐ Think: £ + £ + £ = 1.
Math 102, Intermediate Algebra Section 4.4
Page 20 of 39
Demonstration Problems Practice Problems Write with rational exponents
1. (a) 6 2. (a) 53 3. (a) x5
4. (a) 1x3
Simplify
1. (b) 5 2. (b) 104 3. (b) x53
4. (b) 1a5
Answers: 1. (b) 512 ; 2. (b) 10
14 ; 3. (b) x
53 ; 4. (b) a− 1
5
Math 102, Intermediate Algebra Section 4.4
Page 21 of 39
Example (c) Simplify 1634 .
1634 = 24( )
34
= 24 •
34
= 23 = 8
Example (d) Simplify 16−
34 .
16−
34 = 1
1634
= 18
Example (e) Write 534 in radical notation.
534 = 534
Demonstration Problems Practice Problems Simplify.
5. (a) 2713
6. (a) 164
⎛⎝⎜
⎞⎠⎟
13
Simplify.
5. (b) 3215
6. (b) 164
⎛⎝⎜
⎞⎠⎟
12
Answers: 5. (b) 2; 6. (b) 18
Math 102, Intermediate Algebra Section 4.4
Page 22 of 39
Demonstration Problems Practice Problems
Simplify.
7. (a) 6423
8. (a) y4( )34
Write in radical notation
9. (a) y34
Simplify.
7. (b) 1632
8. (b) a3( )53
Write in radical notation
9. (b) (ab)14
Answers: 7. (b) 64; 8. (b) a5; 9. (b) ab4
Example (f ) Use properties of exponents to simplify x8( )12 .
x8( )12 = x
8 • 12
= x4
Example (g) Use properties of exponents to simplify x14 • x
− 17 .
x14 • x
− 17 = x
14
− 17
= x7
28 −
428
= x3
28
Math 102, Intermediate Algebra Section 4.4
Page 23 of 39
Demonstration Problems Practice Problems Use properties of exponents to simplify
10. (a) x5y10( )15
11. (a) x34 x
54
12. (a) x− 34 x
54
13. (a) x12
x14
Use properties of exponents to simplify
10. (b) w2z6( )12
11. (b) a35a
25
12. (b) a−25a
35
13. (b) a23
a13
Answer: 10. (b) wz3; 11. (b) a; 12. (b) a15 ; 13. (b) a
13
Math 102, Intermediate Algebra Section 4.4
Page 24 of 39
Example (h) Use rational exponents to simplify 86 .
86 = 236
= 236
= 212
= 2
Example (i) Use properties of exponents to simplify 49x24 .
49x24 = 72 x24
= 72 x2( )14
= 72•14 • x
2•14
= 712 • x
12
= 7 • x
7x
Example (j) Use rational exponents to write 84 • 25 as a single radical.
84 • 25 = 234 • 25
= 234 • 2
15
= 21520
+ 420
= 21920
= 21920
Example (k) Use rational exponents to write x345
as a single radical.
x345 = x3( )14
⎛⎝⎜
⎞⎠⎟
15
= x3 •
14
• 15
= x3
20
Math 102, Intermediate Algebra Section 4.4
Page 25 of 39
Demonstration Problems Practice Problems
Use rational exponents to simplify 14. (a) 25x8 15. (a) 16x4y124 Use rational exponents to write each expression as a single radical
16. (a) x23 • x
17. (a) x33
Use rational exponents to simplify 14. (b) 49a4 15. (b) 8x6y123
Use rational exponents to write each expression as a single radical
16. (b) a23 • a5
17. (b) a45
Answer: 14. (b) 7a2; 15. (b) 2x2y4; 16. (b) a1315 ; 17. (b) a25
Math 102, Intermediate Algebra Section 4.5
Page 26 of 39
4.5 Solving Equations with Radicals
Let’s explore.
Suppose x = 4 .
What number has the square root, 4?
We know 16 = 4 .
So then x = 16.
We could have solved this equation using the Power Rule for Solving Equations.
For any real numbers, a, b, and n, such that an is a real number,
if a = b, then an = bn
With this rule, we could have solved the first equation as follows:
x = 4
x( )2 = 42
x( ) x( ) = 16
x2( ) = 16
x = 16
In general, to solve an equation that contains radicals,
Step 1: Isolate the radical.
Step 2: Apply the power rule.
Step 3: Solve the resulting equation. If it still contains a radical repeat steps 1 and 2.
Step 4: Check all solutions for extraneous solutions.
Math 102, Intermediate Algebra Section 4.5
Page 27 of 39
Example (a) Solve 2x −1 = 9
Example (b) Solve 2x + 8 − x = 0
2x + 8 − x = 0 +x +x 2x + 8 + 0 = x
2x + 8( )2 = x2
2x + 8 = x2 –2x –2x 0 + 8 = x2 – 2x –8 –8 0 = x2 – 2x – 8 0 = (x – 4)(x + 2) x – 4 = 0 or x + 2 = 0 +4 +4 –2 –2 x + 0 = 4 x + 0 = –2 x = 4 or x = –2
Example (c) Solve 5x − 23 = 2
2x – 1 = 81 +1 +1 2x + 0 = 82
x = 41
Check 4:
?
?
?
?
0 = 0 Yes
Check –2:
?
?
?
? 4 = 0? No
Solution set: {4}
Solution set: {41}
Check 41:
?
?
? Yes.
Solution set: {2}
5x – 2 = 8 +2 +2 5x + 0 = 10
x = 2
Check 2:
?
?
? Yes.
Math 102, Intermediate Algebra Section 4.5
Page 28 of 39
Example (d) Solve 5x + 6 + 3x + 4 = 2
5x + 6 + 3x + 4 = 2
− 3x + 4 − 3x + 4 5x + 6 + 0 = 2 − 3x + 4
5x + 6( )2 = 2 − 3x + 4( )2
5x + 6 = 2 − 3x + 4( ) 2 − 3x + 4( )
5x + 6 = 4 − 2 3x + 4 − 2 3x + 4 + 3x + 4( )2
5x + 6 = 4 − 4 3x + 4 + 3x + 4 5x + 6 = −4 3x + 4 + 3x + 8 –3x –3x 2x + 6 = −4 3x + 4 + 0 + 8 –8 –8 2x − 2 = −4 3x + 4 + 0
2x − 2( )2 = −4 3x + 4( )2
2x − 2( ) 2x − 2( ) = −4( )2 3x + 4( )2
4x2 – 4x – 4x + 4 = 16(3x + 4) 4x2 – 8x + 4 = 48x + 64 –48x –48x 4x2 – 56x + 4 = 0 + 64 –64 –64 4x2 – 56x – 60 = 0 4(x2 – 14x – 15) = 0 4(x – 15)(x + 1) = 0
4(x −15)(x +1)4
= 04
(x – 15)(x + 1) = 0 x – 15 = 0 or x + 1 = 0
+15 +15 –1 –1 x + 0 = 15 or x + 0 = –1
x = 15 or x = –1
Check 15:
? ? ? 9 + 7 = 2? No
Check –1:
?
? ? 1 + 1 = 2? Yes
Solution set: {–1}
Math 102, Intermediate Algebra Section 4.5
Page 29 of 39
Demonstration Problems Practice Problems Solve.
1. (a) x + 2 + 2 = 5 2. (a) x +13 = 1 3. (a) 12 − x = x
Solve.
1. (b) x − 3 +1= 4 2. (b) x − 23 = 2 3. (b) 4x + 5 = x
Answers: 1. (b) {12}; 2. (b) {10}; 3. (b) {5}
Math 102, Intermediate Algebra Section 4.5
Page 30 of 39
Demonstration Problems Practice Problems Simplify. 4. (a) x − 4 = x − 2 5. (a) 3x + 4 = −5
Simplify. 4. (b) x + 3 = x +1 5. (b) 2x + 5 = −1
Answers: 4. (b) {1}; 5. (b) ∅
Math 102, Intermediate Algebra Section 4.5
Page 31 of 39
Pythagorean Theorem
By the Pythagorean Theorem, in a right triangle, the sum of the squares of the legs is equal to the square of the hypotenuse. That is,
a2 + b2 = c2
We can use this theorem to find the lengths of unknown sides in a right triangle, when two side lengths are given.
Example (e) Find the length of the unknown side of the right triangle.
By the Pythagorean Theorem, we have that a2 + b2 = c2
a2 + 102 = 202
a2 + 100 = 400
–100 –100 a2 + 0 = 300
a2 = 300
a = 300
a = 100 • 3
a = 10 3
or a ≈ 17.3
10
20
10
20
The length of the unknown side is approximately 17.3
Math 102, Intermediate Algebra Section 4.5
Page 32 of 39
Demonstration Problems Practice Problems 6. (a) Find the length of the unknown side.
6. (b) Find the length of the unknown side.
Answers: 6. (b) 2 11
10
12
7
9
Math 102, Intermediate Algebra Section 4.6
Page 33 of 39
4.6 Complex Numbers
Common Number Sets
Symbol Name Description Notation or examples
Natural
Numbers The counting numbers beginning with 1. = {1, 2, 3, …}
Whole Numbers The counting numbers and zero. {0, 1, 2, 3, …}
Integers The whole numbers and their opposites.
={… –3, –2, –1, 0, 1, 2, 3, …} Z comes from the German word for number, zahlen, since I is used for Imaginary Numbers.
Rational Numbers
Numbers of the form ab
where a is
an integer and b is a nonzero integer.
Examples: 13
, 10.5, –132
, 12, 4
Q is for quotient, since R is used for Real Numbers.
Irrational Numbers
Real numbers that are not rational numbers.
Examples: π, 2 , 5 15
Real
Numbers All rational and irrational numbers. All of the above.
At the right is a Venn Diagram of the number sets from above. The diagram shows that natural numbers are contained in whole numbers which are contained in the integers, which are contained in the rational numbers. The irrational numbers are disjoint from the rational numbers. The real numbers are made up of both the rational and irrational numbers.
Until now, we have been using these number sets to solve equations. But consider the equation
x2 = − 4 What number has the square, – 4?
We know 22 = 4 and (–2)2 = 4. But no real number squared equals – 4.
Using real numbers only the solution set to this equation is empty.
Math 102, Intermediate Algebra Section 4.6
Page 34 of 39
An Italian mathematician Gerolamo Cardan (1501, 1576) was curious about solutions to cubic polynomial equations of the form ax3 + bx + c = 0. Determined not to let the problem of square roots of negative numbers stop him, he developed the concept of −1 . Today, complex numbers have real world applications in many fields, two of which are electronics and electrical engineering.
We define
i = −1
Complex numbers = {a + bi | a ∈ , b ∈ , and i = −1 }. When b = 0, then the complex number becomes a real number. Thus, the complex numbers include all of the real numbers.
We can now extend
the number set Venn Diagram to
Example (a) Write −9 as a product of a real number and i.
−9 = −1• 9 = −1 • 9 = i • 3 = 3i
Example (b) Write −20 as a product of a real number and i.
−20 = −1• 22 • 5 = −1 • 22 • 5 = i • 2 • 5 = 2i 5
Math 102, Intermediate Algebra Section 4.6
Page 35 of 39
Demonstration Problems Practice Problems
Write as a product of a real number and i.
1. (a) −36 2. (a) −40 Assume x ∈ .
3. (a) −x6
Write as a product of a real number and i.
1. (b) −49 2. (b) −50 Assume x ∈ .
3. (b) −x8
Answers: 1. (b) 7i; 2. (b) 5i 2 ; 3. (b) x4i
Math 102, Intermediate Algebra Section 4.6
Page 36 of 39
Example (c) Simplify (2 + 3i) + (5 – 4i). Write your answer in the form a + bi. (2 + 3i) + (5 – 4i) = 2 + 3i + 5 + –4i = 2 + 5 + 3i + –4i = 7 + –i = 7 – i Since i = −1 , then i2 = −1( )2 = −1 .
Example (d) Simplify (2 + 3i)(5 – 4i). Write your answer in the form a + bi. (2 + 3i)(5 – 4i) = 10 + –8i + 15i + –12i2 = 10 + 7i + –12(–1) = 10 + 7i + 12 = 22 + 7i
Demonstration Problems Practice Problems Simplify. 4. (a) (5 + 2i) – (3 – i) 5. (a) (1 + 4i)(1 – 4i)
Simplify. 4. (b) (4 + 6i) – (2 – 3i) 5. (b) (2 + 3i)(2 – 3i)
Answers: 4. (b) 2 + 9i; 5. (b) 13
Math 102, Intermediate Algebra Section 4.6
Page 37 of 39
Example (e) Rationalize the denominator and simplify 35i
.
35i
= 35i• ii
= 3i5i2
= 3i5(−1)
= 3i−5
= − 35i
Example (f ) Rationalize the denominator and simplify 13+ i
.
Write your answer in the form a + bi.
13+ i
= 13+ i
• 3− i3− i
= 1(3− i)(3+ i)(3− i)
= 3− i9 − 3i + 3i − i2
= 3− i9 − (−1)
= 3− i10
= 310
− 110i
Math 102, Intermediate Algebra Section 4.6
Page 38 of 39
Demonstration Problems Practice Problems Rationalize the denominator and simplify. Write your answer in the form a + bi.
6. (a) 13i
7. (a) 1+ 2i1− 2i
Rationalize the denominator and simplify. Write your answer in the form a + bi.
6. (b) 25i
7. (b) 1+ 2i1+ 3i
Answers: 6. (b) −25i ; 7. (b)
710
− 110i
Math 102, Intermediate Algebra Section 4.6
Page 39 of 39
Powers of i Complete the following:
i = i
i2 = –1
i3 = i2 • i = –1 • i = –i
i4 = i2 • i2 = (–1)(–1) = 1
i5 = i4 • i = = i
i6 = i2 • i2 • i2 = =
i7 = i6 • i = =
i8 = i2 • i2 • i2 • i2 = =
i9 = i8 • i = =
i10 = i2 • i2 • i2 • i2 • i2 = =
Demonstration Problems Practice Problems
Simplify 8. (a) i12 = 9. (a) i85 =
Simplify 8. (b) i15
9. (b) i90 =
Answers: 8. (b) –i; 9. (b) –1
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