Download Chapter 7 Note Packet

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_
Zero and Negative Exponents
Zero as an Exponent:
For every nonzero number a, a° =1.
Examples:
r°
2 3L
-‘
Negative Exponent:
For every nonzero number a and integer n, a
=
I
Examples:
Example 1: Simplifying a Power
Simplify.
a.
43
b.
(—l.23)°
Example 2: Simplifying an Exponential Expression
Simplify each expression.
b.-i
Example 3: Evaluating an Exponential Expression
Evaluate 3m
t for m
2
=
2 and t = -3.
37
12
3
Multiplication Properties of Exponents
Mu1tiplyin Powers With the Same Base Property:
For every nonzero number a and integers m and n, am a”
Examples: 5)_
5
2
TLi
=
51
.
m
a
2’
Exampie 1: Multiplying Powers
Rewrite each expression using each base only once.
a. ii
4 •ii
3
b.
(
52 .52
5
‘
2 2
-
[1
Example 2: Multiplying Powers in an Algebraic Expression
Simplify each expression.
a.
5 3n
2n
2
b. 5x2y
4 •3x
8
4
°LN
Raising a Power to a Power Property:
For every nonzero number a and integers m and n, (a
)”
tm
Examples:
2..\)3
5-
=
t”.
a
m
(—2’)
Example 3: Simplifying a Power Raised to a Power
Simplify 3
(x )6
2
Example 4: Simplifying an Expression With Powers
Simplify c’(c
.
2
)
3
Li
Raising a Product to a Power Property:
For every nonzero number a and b and integer n, (ab)’
7
=
(3\)a.
Examples:
Example 5: Simplifying a Product Raised to a Power
Write the expression that represents the area of the square.
2x
-.
tL
(axjz
2
Example 6: Simplifying a Product Raised to a Power
(3xy
(x
.
4
)
Simplify 2
N,
cHI1f
,J:71
3
a’.
32.
2
92
Division Properties of Exponents
Dividing Powers With the Same Base Property:
For every nonzero number a and integers m and n,
Example:
=
.
3
a’
2 2
Example 1: Simplifying an Algebraic Expression
Simplify each expression.
3
d
1
c
-2
a.
ft.
L
1
Cd
Raising a Ouotient to a Power Property:
For every nonzero number a and b and integer n(J
Example:
(
=
/
Example 2: Raising a Quotient to a Power
(4N3
Which expression is equivalent to I —i- ?
x)
3
H
b
1
r
(L
-
4
Example 3: Simplifying an Exponential Expression
Simplify each expression.
a. H—
—
—
5
___
_________
____________
______________
Section 7.1
Since
52
Since
53
Since
54
Since
55
=
25, 5 is a
=
125, 5 is a
Roots and Radical Expressions
root of 25.
root of 125.
rooto
=625,5isaC(+k
rootof
=3125,5isa
3j
Definition of the nth Root:
For any real numbers a and b, and any positive integer n, if a’
root of b.
oc
9
-
=
c*d
b, then a is an nth
—
O
4- rcc*s
)-I(hC5flOftoi
-
4Z
khQ C
‘
j
Summary of the possible real roots of a real number.
Type of Nimiber
Number of Rea’ nth
Roots When n s Even
NLInbef’ of ReI ,,th
ROGtS Weii IF Js Odd
-:JciF(\)
c
QOc h
ñ
Example 1: Finding All Real Roots
Find all the real roots.
a. The cube roots of 8, -1000, and
I
hQ1\cc4
t\
-DC
\‘\
hQ\’Q_I OO&
b. The fourth roots of 1, -0.0001, and
L
‘jiI.t
0
Iäo
16
—.
c{th rooi-
od
V 1T) hCJL QJJ
--
,
(TJ3
-
S
VH and
1JTD1O
-
I
1
5
b
1O
(-r)Z
I
&t
V
\,cJ —ko
Radical Sign:
Radicand:
Index:
\r$&CCk(
cctr rdr * tijç oJ
—[-
Th°
PrincipalRoot:
TYL
a
-.
n c n’JU)cr
-JQ
i
4 rc
nc
ThQ
Lth
O
rc’ct
\CkS
_codcO
+o oc*.
6
Q$
T
Example 2: Finding Roots
Find each real-number root.
a.Vi
b.
c@oJ iu&br
‘S
Sc)c\m 3
-;
Example 3: Simplifying Radical Expressions
Simplify each radical expression.
a.
b.
Ja3b6
C.
Jx’y
J—ioo
-
5T::
jL
2
__
Section 7.2
Multiplyin2 and Dividing Radical Expressions
MultipIyin Radical Expressions Property:
If
and ‘[ are real numbers, then
.
=
Example 1: Multiplying Radicals
Multiply. Simplify if possible.
-\
=
.
\-
pap+j cbQS nc* cppk
S rob- a QcJ nuiDg.
scc
Example 2: Simplifying Radical Expressions
Simplify each expression. Assume that all variables are positive. Then absolute value
symbols are never needed in the simplified expression.
a.
b.
\J
kI8On
z
1
J
0
)ion
Dividing Radical Expressions Property:
If and arerealnumbersand bO,then
3
___
Example 4: Dividing Radicals
Divide and simplify. Assume that all variables are positive.
a.
3
b
Rationalize the Denominator
()Q
O
f)J ç-
\Dd cc.
-\-\Q-
áQf1ofl ncO1
Example 5: Rationalizing the Denominator
Rationalize the denominator of each expression. Assume that all variables are positive.
a
•J
_1çI
\{3
4D
‘I
b
-
5X\/
—
c.
3J_
V 3x
-
.L’E
c
L_
4
Section 7.3
Like Radicals:
-
Binomial Radical Expressions
ea\ ftSS
[CC
-W’&... bxiQ
ubtingdial
Example!: Adig
Add or subtract if possible.
a. 5k1—3&
b.
4Ii+5J
rcC-
Example 2: Simplifying Before Adding and Subtracting
Simplify 6-lu
+
4J
—
3-In.
J9Y -33
4l_
-L:
+
Example 3: Multiplying Binomial Radical Expressions
Multiply (3+ 2J)(2 + 4J).
Pci :
t
4
Ljoj
5
Example 4: Multiplying Conjugates
Multiply (2+ J)(2
—
Fo IL cc
QT
DrQnc c
(cbbcb2
-
()z
Example 5: Rationalizing Binomial Radical Denominators
Rationalize the denominator of
3+
4
(3
-
I
-
‘-5
-Lj
6
\
Section 7.4
Rational Exponents
Rational Exponent:
CO
-
1sScor.
VL
Example 1: Simplifying Expressions With Rational Exponents
Simplify each expression.
a.
125
b.
5 5}
1o.1oo
5
\)
C)
=
Definition of Rational Exponents:
If the nth root of a is a real number and m is an integer, then
and a =
a=
-
TL
(
‘J
Example 2: Converting to and From Radical Form
5 in radical form.
.
2
a. Write the exponential expressions x and y
b. Write the radical expressions
J1 and
(J)2
in exponential form.
2
b
(Th(
OCrdk
Summary of Properties of Rational Exponents:
Let m and n represent rational numbers. Assume that no denominator equals 0.
Example
Property
(
4
]‘Lg
am+n
1’1 a
a
(am)n =am
-
1
=a
1
n
11 11
(ab)
a _n
L3
)2..
-
5
1)
9Vi
a”
m
=
L
11
a
f
(a
\11
“
[L’\3
a
11
b
3
ai
Example 4: Simplifying Numbers With Rational Exponents
Simplify each number.
a.
=
_33/5
435
b.
(—32)
3
\j(33)
L
.
-
L4
N
I
ExampleS: Writing Expressions in Simplest Form
Write (16y
) in simplest form.
8
1(s”I
J4/
1
\
—_i-—
\j
Ni
1
)
3
8
.-
Section 7.5
Solve Square Root and Other Radical Equations
Radical Equation: ç0
rn
Solving a Radical Equation.
c+- +
\xI) @ ‘\Jc
c
sc tcU
cjxthc
+hp rod
-
cc]
thSQ
c
boW’
QE*
or Sc&
c$
Qa\ O
Exampie 1: Solving Square Root Equations
-
Solve 2+iJ3x—2=6.
\3K-
H
CHY
-i-•
+i4
-
Example 2: Solving Radical Equations With Rational Exponents
Solve 2(x—2) =50.
X-
5Q.
53
tb
(x
x
CbQC.
0
cj9
r’3
3Q
-
::
50
‘A
-
V
I’
-.
1 ‘F
cNc
.‘
50
x-.--. .(ES
x\)
:
I
•(
9
50
50
50
53DQ
Extraneous Solutions:
•i
Qd om
A tuEcn d cn
c± cn
Example 3: Checking For Extraneous Solutions
Solve -Jx —3 +5
=
x. Check
5K
for extraneous solutions.
(7S
-5
(i-’Xx
Al)
7
\rTE ÷31
A
J+ 5
f5
32jj
.3
\J
7
÷5
—H
-4-3
_o
1 7 ,/
Example 4: Solving Equations With Two Rational Exponents
Solve (2x + 1)0.5
—
(3x + 4)0.25
(3KH)
0. Check for extraneous solutions.
ChQch
(./) + O05 (3(/)
/c
.
/3/
17 ÷
c
\
(±i)
L
(%)°
z(*t)
/
q
q
4
+(
-1-
-.
L
.
-
So
-,
--
(i:)
(E:) Q3
3x4
q
)
(3
iEz
5
oc -1
4xx -3
1
o
0
--
(x i)a
D(S
.7
0
--
2
f
I
\Q.5
v-I)
rai
(4K-3jy o
10
/
“I)
±0
i-um.br-