Section 7.1
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![Page 1: Section 7.1](https://reader030.fdocuments.in/reader030/viewer/2022032612/568131be550346895d9827fd/html5/thumbnails/1.jpg)
Section 7.1
Rational Exponents and Radicals
![Page 2: Section 7.1](https://reader030.fdocuments.in/reader030/viewer/2022032612/568131be550346895d9827fd/html5/thumbnails/2.jpg)
OBJECTIVES
A Find the nth root of a number, if it exists.
![Page 3: Section 7.1](https://reader030.fdocuments.in/reader030/viewer/2022032612/568131be550346895d9827fd/html5/thumbnails/3.jpg)
OBJECTIVES
B Evaluate expressions containing rational exponents.
![Page 4: Section 7.1](https://reader030.fdocuments.in/reader030/viewer/2022032612/568131be550346895d9827fd/html5/thumbnails/4.jpg)
OBJECTIVES
C Simplify expressions involving rational exponents.
![Page 5: Section 7.1](https://reader030.fdocuments.in/reader030/viewer/2022032612/568131be550346895d9827fd/html5/thumbnails/5.jpg)
DEFINITION
If a and x are real numbers and n is a positive integer:
x is an nth root of a if xn = a
NTH ROOT
![Page 6: Section 7.1](https://reader030.fdocuments.in/reader030/viewer/2022032612/568131be550346895d9827fd/html5/thumbnails/6.jpg)
DEFINITIONPRINCIPLE NTH ROOT
If n is a positive integer, then an
denotes the principle nth root of a
![Page 7: Section 7.1](https://reader030.fdocuments.in/reader030/viewer/2022032612/568131be550346895d9827fd/html5/thumbnails/7.jpg)
DEFINITIONRATIONAL EXPONENTS AND THEIR ROOTS
a1/ n = an
If n is a positive integer andan is a real number:
![Page 8: Section 7.1](https://reader030.fdocuments.in/reader030/viewer/2022032612/568131be550346895d9827fd/html5/thumbnails/8.jpg)
DEFINITIONRADICAL EXPRESSION WITH AN M/N EXPONENT
am/ n = ( an )m = amn
Provided m and n are positive integers and an
is a real number.
![Page 9: Section 7.1](https://reader030.fdocuments.in/reader030/viewer/2022032612/568131be550346895d9827fd/html5/thumbnails/9.jpg)
LAWS OF EXPONENTS
I. r s r+s a a = a
If r, s, and t are rational, and a and b are real:
II.
ar
as = ar–s
![Page 10: Section 7.1](https://reader030.fdocuments.in/reader030/viewer/2022032612/568131be550346895d9827fd/html5/thumbnails/10.jpg)
LAWS OF EXPONENTS
III. r s r s (a ) = a
IV. (ar bs )t = artbst
If r, s, and t are rational, and a and b are real:
![Page 11: Section 7.1](https://reader030.fdocuments.in/reader030/viewer/2022032612/568131be550346895d9827fd/html5/thumbnails/11.jpg)
Practice Test
Exercise #1
Chapter 7Section 7.1A
![Page 12: Section 7.1](https://reader030.fdocuments.in/reader030/viewer/2022032612/568131be550346895d9827fd/html5/thumbnails/12.jpg)
Find, if possible.
a. 3
–64
= – 4
3 3= –4
![Page 13: Section 7.1](https://reader030.fdocuments.in/reader030/viewer/2022032612/568131be550346895d9827fd/html5/thumbnails/13.jpg)
b. –36
It is not a real number.
Find, if possible.
![Page 14: Section 7.1](https://reader030.fdocuments.in/reader030/viewer/2022032612/568131be550346895d9827fd/html5/thumbnails/14.jpg)
Practice Test
Exercise #4
Chapter 7Section 7.1B
![Page 15: Section 7.1](https://reader030.fdocuments.in/reader030/viewer/2022032612/568131be550346895d9827fd/html5/thumbnails/15.jpg)
a. –27 –
23
2
3
1=
27–
Evaluate if possible.
23
1=
27–
2
1=
–3 =
19
![Page 16: Section 7.1](https://reader030.fdocuments.in/reader030/viewer/2022032612/568131be550346895d9827fd/html5/thumbnails/16.jpg)
b. 8– 2
3
= 1
823
= 1
83
2
=
1
22 =
14
Evaluate, if possible.
![Page 17: Section 7.1](https://reader030.fdocuments.in/reader030/viewer/2022032612/568131be550346895d9827fd/html5/thumbnails/17.jpg)
Section 7.2
Simplifying Radicals
![Page 18: Section 7.1](https://reader030.fdocuments.in/reader030/viewer/2022032612/568131be550346895d9827fd/html5/thumbnails/18.jpg)
OBJECTIVES
A Simplify radical expressions.
![Page 19: Section 7.1](https://reader030.fdocuments.in/reader030/viewer/2022032612/568131be550346895d9827fd/html5/thumbnails/19.jpg)
OBJECTIVES
B Rationalize the denominator of a fraction.
![Page 20: Section 7.1](https://reader030.fdocuments.in/reader030/viewer/2022032612/568131be550346895d9827fd/html5/thumbnails/20.jpg)
OBJECTIVES
C Reduce the order of a radical expression.
![Page 21: Section 7.1](https://reader030.fdocuments.in/reader030/viewer/2022032612/568131be550346895d9827fd/html5/thumbnails/21.jpg)
DEFINITIONnTH ROOT
( 0)1/n na = a a
when n is a positive integer.
![Page 22: Section 7.1](https://reader030.fdocuments.in/reader030/viewer/2022032612/568131be550346895d9827fd/html5/thumbnails/22.jpg)
LAWS
I. ( 0)n na = a a
For Simplifying Radical Expressions
![Page 23: Section 7.1](https://reader030.fdocuments.in/reader030/viewer/2022032612/568131be550346895d9827fd/html5/thumbnails/23.jpg)
( 0)Product ruII. le n n n a, b ab = a b
LAWSFor Simplifying Radical Expressions
![Page 24: Section 7.1](https://reader030.fdocuments.in/reader030/viewer/2022032612/568131be550346895d9827fd/html5/thumbnails/24.jpg)
( 0 > 0)Quotient rulIII. e n
nn
a ,b a a
= b b
LAWSFor Simplifying Radical Expressions
![Page 25: Section 7.1](https://reader030.fdocuments.in/reader030/viewer/2022032612/568131be550346895d9827fd/html5/thumbnails/25.jpg)
DEFINITION
ann = | a |
ann
n is an even positive integer
![Page 26: Section 7.1](https://reader030.fdocuments.in/reader030/viewer/2022032612/568131be550346895d9827fd/html5/thumbnails/26.jpg)
DEFINITION ann
n is an odd positive integer
ann = a
![Page 27: Section 7.1](https://reader030.fdocuments.in/reader030/viewer/2022032612/568131be550346895d9827fd/html5/thumbnails/27.jpg)
Practice Test
Exercise #7b
Chapter 7Section 7.2A
![Page 28: Section 7.1](https://reader030.fdocuments.in/reader030/viewer/2022032612/568131be550346895d9827fd/html5/thumbnails/28.jpg)
Simplify.
44 –b. x
= – x
44
= – x
= x
![Page 29: Section 7.1](https://reader030.fdocuments.in/reader030/viewer/2022032612/568131be550346895d9827fd/html5/thumbnails/29.jpg)
Practice Test
Exercise #8b
Chapter 7Section 7.2A
![Page 30: Section 7.1](https://reader030.fdocuments.in/reader030/viewer/2022032612/568131be550346895d9827fd/html5/thumbnails/30.jpg)
Simplify.
b. 3
54a4b12
= 3ab4 3 2a
= 3
27 • 2 • a3 • a • b12
= 3
33 a3 b12 • 2a
![Page 31: Section 7.1](https://reader030.fdocuments.in/reader030/viewer/2022032612/568131be550346895d9827fd/html5/thumbnails/31.jpg)
Practice Test
Exercise #9b
Chapter 7Section 7.2A
![Page 32: Section 7.1](https://reader030.fdocuments.in/reader030/viewer/2022032612/568131be550346895d9827fd/html5/thumbnails/32.jpg)
Simplify.
b. 3 5
x6
= 3
53
x6
=
35
x 2
![Page 33: Section 7.1](https://reader030.fdocuments.in/reader030/viewer/2022032612/568131be550346895d9827fd/html5/thumbnails/33.jpg)
Section 7.3
Operations with Radicals
![Page 34: Section 7.1](https://reader030.fdocuments.in/reader030/viewer/2022032612/568131be550346895d9827fd/html5/thumbnails/34.jpg)
OBJECTIVES
A Add and subtract similar radical expressions.
![Page 35: Section 7.1](https://reader030.fdocuments.in/reader030/viewer/2022032612/568131be550346895d9827fd/html5/thumbnails/35.jpg)
OBJECTIVES
B Multiply and divide radical expressions.
![Page 36: Section 7.1](https://reader030.fdocuments.in/reader030/viewer/2022032612/568131be550346895d9827fd/html5/thumbnails/36.jpg)
OBJECTIVES
C Rationalize the denominators of radical expressions involving sums or differences.
![Page 37: Section 7.1](https://reader030.fdocuments.in/reader030/viewer/2022032612/568131be550346895d9827fd/html5/thumbnails/37.jpg)
DEFINITION
Radical expressions with the same index and the same radicand.
LIKE RADICAL EXPRESSIONS
![Page 38: Section 7.1](https://reader030.fdocuments.in/reader030/viewer/2022032612/568131be550346895d9827fd/html5/thumbnails/38.jpg)
DEFINITION
The expressions a + b and a – b are conjugates of each other.
CONJUGATE
![Page 39: Section 7.1](https://reader030.fdocuments.in/reader030/viewer/2022032612/568131be550346895d9827fd/html5/thumbnails/39.jpg)
Practice Test
Exercise #14
Chapter 7Section 7.3A
![Page 40: Section 7.1](https://reader030.fdocuments.in/reader030/viewer/2022032612/568131be550346895d9827fd/html5/thumbnails/40.jpg)
Perform the indicated operations.
a. 32 + 98
= 16 • 2 + 49 • 2
= 4 2 + 7 2
= 11 2
![Page 41: Section 7.1](https://reader030.fdocuments.in/reader030/viewer/2022032612/568131be550346895d9827fd/html5/thumbnails/41.jpg)
b. 112 – 28
= 16 • 7 – 4 • 7
= 4 7 – 2 7
= 2 7
Perform the indicated operations.
![Page 42: Section 7.1](https://reader030.fdocuments.in/reader030/viewer/2022032612/568131be550346895d9827fd/html5/thumbnails/42.jpg)
Practice Test
Exercise #16b
Chapter 7Section 7.3B
![Page 43: Section 7.1](https://reader030.fdocuments.in/reader030/viewer/2022032612/568131be550346895d9827fd/html5/thumbnails/43.jpg)
Perform the indicated operations.
= 3
3x •3
9x 2 –3
3x •3
16x
= 3
27x3 –3
48x 2
= 3x –3
8 • 6x 2
= 3x – 23
6x 2
333 2b. 3 9 – 16
x x x
![Page 44: Section 7.1](https://reader030.fdocuments.in/reader030/viewer/2022032612/568131be550346895d9827fd/html5/thumbnails/44.jpg)
Practice Test
Exercise #18b
Chapter 7Section 7.3B
![Page 45: Section 7.1](https://reader030.fdocuments.in/reader030/viewer/2022032612/568131be550346895d9827fd/html5/thumbnails/45.jpg)
Find the product.
– 6 b + 3 6. 3
–= 6 + 3 6 3
2 2–= 6 3
= 6 – 3
(a + b)(a – b) = a2 – b2
= 3
![Page 46: Section 7.1](https://reader030.fdocuments.in/reader030/viewer/2022032612/568131be550346895d9827fd/html5/thumbnails/46.jpg)
Practice Test
Exercise #20
Chapter 7Section 7.3C
![Page 47: Section 7.1](https://reader030.fdocuments.in/reader030/viewer/2022032612/568131be550346895d9827fd/html5/thumbnails/47.jpg)
Rationalize the denominator.
2
x – 5
=
2
x – 5 •
x + 5
x + 5
=
2 ( x + 5)
x – 25
=
2 x + 10x – 25
![Page 48: Section 7.1](https://reader030.fdocuments.in/reader030/viewer/2022032612/568131be550346895d9827fd/html5/thumbnails/48.jpg)
Section 7.4
Solving Equations Containing Radicals
![Page 49: Section 7.1](https://reader030.fdocuments.in/reader030/viewer/2022032612/568131be550346895d9827fd/html5/thumbnails/49.jpg)
OBJECTIVES
A Solve equations involving radicals.
![Page 50: Section 7.1](https://reader030.fdocuments.in/reader030/viewer/2022032612/568131be550346895d9827fd/html5/thumbnails/50.jpg)
OBJECTIVES
B Solve applications requiring the solution of radical equations.
![Page 51: Section 7.1](https://reader030.fdocuments.in/reader030/viewer/2022032612/568131be550346895d9827fd/html5/thumbnails/51.jpg)
DEFINITIONPOWER RULE OF EQUATIONS
All solutions of the equation P = Q are solutions of the equation Pn = Qn , where n is a natural number.
![Page 52: Section 7.1](https://reader030.fdocuments.in/reader030/viewer/2022032612/568131be550346895d9827fd/html5/thumbnails/52.jpg)
TO SOLVE EQUATIONS CONTAINING RADICALS
PROCEDURE
• Isolate
• Raise
• Simplify
• Repeat
• Solve
• Check
![Page 53: Section 7.1](https://reader030.fdocuments.in/reader030/viewer/2022032612/568131be550346895d9827fd/html5/thumbnails/53.jpg)
Practice Test
Exercise #21
Chapter 7Section 7.4A
![Page 54: Section 7.1](https://reader030.fdocuments.in/reader030/viewer/2022032612/568131be550346895d9827fd/html5/thumbnails/54.jpg)
There is no real number solution
because x + 2 ° negative number.
Solve.
a. x + 2 = – 2
When solving by squaring both sides:x + 2 = 4
x = 2
However, the solution x = 2 does not check.
![Page 55: Section 7.1](https://reader030.fdocuments.in/reader030/viewer/2022032612/568131be550346895d9827fd/html5/thumbnails/55.jpg)
Solve.
b. x + 2 = x – 10
2 2 + 2 = – 10x x
x + 2 = x 2 – 20x + 100
0 = x 2 – 21x + 98
0 = – 14 – 7x x
x – 14 = 0 oror x – 7 = 0
x = 14 x = 7
![Page 56: Section 7.1](https://reader030.fdocuments.in/reader030/viewer/2022032612/568131be550346895d9827fd/html5/thumbnails/56.jpg)
Solve. x = 14 x = 7
Check:
9 ° – 3
7 + 2 = ?
7 – 10
14 + 2 = ?
14 – 10
16 = 4
The only solution is x = 14 .
b. x + 2 = x – 10
x = 7,
x = 14,
![Page 57: Section 7.1](https://reader030.fdocuments.in/reader030/viewer/2022032612/568131be550346895d9827fd/html5/thumbnails/57.jpg)
Practice Test
Exercise #22
Chapter 7Section 7.4A
![Page 58: Section 7.1](https://reader030.fdocuments.in/reader030/viewer/2022032612/568131be550346895d9827fd/html5/thumbnails/58.jpg)
x – 3 – x = – 3
x – 3 = x 2 – 6x + 9
x – 3 = x – 3
2 2
– 3 = – 3x x
–3 = x 2 – 7x + 9
Solve.
![Page 59: Section 7.1](https://reader030.fdocuments.in/reader030/viewer/2022032612/568131be550346895d9827fd/html5/thumbnails/59.jpg)
0 = – 4 – 3x x
x – 4 = 0 oror x – 3 = 0
0 = x 2 – 7x + 12
–3 = x 2 – 7x + 9
x = 4 x = 3
x – 3 – x = – 3
Solve.
![Page 60: Section 7.1](https://reader030.fdocuments.in/reader030/viewer/2022032612/568131be550346895d9827fd/html5/thumbnails/60.jpg)
x = 4 x = 3
Check:
–3 = – 3
x = 4, 4 – 3 – 4 = ?
– 3
The solutions are x = 4 or x = 3 .
x = 3, 3 – 3 –3 = ?
– 3
–3 = – 3
x – 3 – x = – 3
Solve.
![Page 61: Section 7.1](https://reader030.fdocuments.in/reader030/viewer/2022032612/568131be550346895d9827fd/html5/thumbnails/61.jpg)
Section 7.5
Complex Numbers
![Page 62: Section 7.1](https://reader030.fdocuments.in/reader030/viewer/2022032612/568131be550346895d9827fd/html5/thumbnails/62.jpg)
OBJECTIVES
A Write the square root of a negative integer in terms of i.
![Page 63: Section 7.1](https://reader030.fdocuments.in/reader030/viewer/2022032612/568131be550346895d9827fd/html5/thumbnails/63.jpg)
OBJECTIVES
B Add and subtract complex numbers.
![Page 64: Section 7.1](https://reader030.fdocuments.in/reader030/viewer/2022032612/568131be550346895d9827fd/html5/thumbnails/64.jpg)
OBJECTIVES
C Multiply and divide complex numbers.
![Page 65: Section 7.1](https://reader030.fdocuments.in/reader030/viewer/2022032612/568131be550346895d9827fd/html5/thumbnails/65.jpg)
OBJECTIVES
D Find powers of i.
![Page 66: Section 7.1](https://reader030.fdocuments.in/reader030/viewer/2022032612/568131be550346895d9827fd/html5/thumbnails/66.jpg)
DEFINITIONCOMPLEX NUMBER
a + bi
If a and b are real numbers, the following is a complex number:
Real part Imaginary part
![Page 67: Section 7.1](https://reader030.fdocuments.in/reader030/viewer/2022032612/568131be550346895d9827fd/html5/thumbnails/67.jpg)
RULESADDING AND SUBTRACTING COMPLEX NUMBERS
(a + bi) + (c + di) = (a + c) + (b +d)i
(a + bi) – (c + di) = (a – c) + (b – d)i
![Page 68: Section 7.1](https://reader030.fdocuments.in/reader030/viewer/2022032612/568131be550346895d9827fd/html5/thumbnails/68.jpg)
PROCEDUREDIVIDING ONE COMPLEX NUMBER BY ANOTHER
Multiply the numerator and the denominator by the conjugate of the denominator.
![Page 69: Section 7.1](https://reader030.fdocuments.in/reader030/viewer/2022032612/568131be550346895d9827fd/html5/thumbnails/69.jpg)
Practice Test
Exercise #26b
Chapter 7Section 7.5A
![Page 70: Section 7.1](https://reader030.fdocuments.in/reader030/viewer/2022032612/568131be550346895d9827fd/html5/thumbnails/70.jpg)
Write in terms of i.
b. –98 = –1 • 49 • 2
= i • 7 2
= 7i 2
![Page 71: Section 7.1](https://reader030.fdocuments.in/reader030/viewer/2022032612/568131be550346895d9827fd/html5/thumbnails/71.jpg)
Practice Test
Exercise #27b
Chapter 7Section 7.5B
![Page 72: Section 7.1](https://reader030.fdocuments.in/reader030/viewer/2022032612/568131be550346895d9827fd/html5/thumbnails/72.jpg)
Find.
b. 5 + 2 – 7 – 8i i
= 5 + 2i – 7 + 8i
= – 2 + 10i
= 5 – 7 + 2i + 8i
![Page 73: Section 7.1](https://reader030.fdocuments.in/reader030/viewer/2022032612/568131be550346895d9827fd/html5/thumbnails/73.jpg)
Practice Test
Exercise #29b
Chapter 7Section 7.5C
![Page 74: Section 7.1](https://reader030.fdocuments.in/reader030/viewer/2022032612/568131be550346895d9827fd/html5/thumbnails/74.jpg)
Find.
b.
4 – 3i3 – 5i
3 + 4 – 3 =
3
5
3 + 5 – 5
i i
i i
2 2
4 3 + 4 5 + –3 3 + –3 5 =
3 – 5
i i i i
i
![Page 75: Section 7.1](https://reader030.fdocuments.in/reader030/viewer/2022032612/568131be550346895d9827fd/html5/thumbnails/75.jpg)
Find.
b.
4 – 3i3 – 5i
2 2
4 3 + 4 5 + –3 3 + –3 5 =
3 – 5
i i i i
i
=
12 + 20i – 9i – 15i 2
9 – 25i 2
![Page 76: Section 7.1](https://reader030.fdocuments.in/reader030/viewer/2022032612/568131be550346895d9827fd/html5/thumbnails/76.jpg)
Find.
b.
4 – 3i3 – 5i
=
12 + 20i – 9i – 15i 2
9 – 25i 2
=
12 + 11i + 159 + 25
=
27 + 11i34
= 2734
+ 1134
i
![Page 77: Section 7.1](https://reader030.fdocuments.in/reader030/viewer/2022032612/568131be550346895d9827fd/html5/thumbnails/77.jpg)
Practice Test
Exercise #30b
Chapter 7Section 7.5D
![Page 78: Section 7.1](https://reader030.fdocuments.in/reader030/viewer/2022032612/568131be550346895d9827fd/html5/thumbnails/78.jpg)
Write the answer as 1, -1, i or -i.
b. i – 9
=
1
i 9 =
1
i8 • i
= 1
i 4 2
• i
=
1i
= 1
1 • i =
1 • ii • i
=
i
i 2 =
i–1
= – i