Exponents and Radicals
Objective: To review rules and properties of exponents and
radicals.
Exponential Notation
Properties of Exponents
Properties of Exponents
Example 1
• Use the properties of exponents to simplify each expression.
a) )4)(3( 34 abab
Example 1
• Use the properties of exponents to simplify each expression.
a) )4)(3( 34 abab
baabab 234 12)4)(3(
Example 1
• Use the properties of exponents to simplify each expression. You Try:
b) 32 )2( xy
Example 1
• Use the properties of exponents to simplify each expression. You Try:
b) 32 )2( xy
63323332 8)(2)2( yxyxxy
Example 1
• Use the properties of exponents to simplify each expression. You Try:
c) 02 )4(3 aa
Example 1
• Use the properties of exponents to simplify each expression. You Try:
c) 02 )4(3 aa
0,3)1(3)4(3 02 aaaaa
Example 1
• Use the properties of exponents to simplify each expression. You Try:
d) 235
y
x
Example 1
• Use the properties of exponents to simplify each expression. You Try:
d) 235
y
x
2
6
2
23223 25)(55
y
x
y
x
y
x
Example 2
• Rewrite each expression with positive exponents.
a) 1x
Example 2
• Rewrite each expression with positive exponents.
a) 1x
xx
11
Example 2
• Rewrite each expression with positive exponents.
b) 23
1x
Example 2
• Rewrite each expression with positive exponents.
b) 23
1x
3
1
3
1
3
1 2
22
x
xx
Example 2
• Rewrite each expression with positive exponents.• You Try:
c) ba
ba2
43
4
12
Example 2
• Rewrite each expression with positive exponents.• You Try:
c) ba
ba2
43
4
12
5
5
4
23
2
43 3
4
12
4
12
b
a
bb
aa
ba
ba
Example 2
• Rewrite each expression with positive exponents.• You Try:
d) 223
y
x
Example 2
• Rewrite each expression with positive exponents.• You Try:
d) 223
y
x
4
2
222
22
2
22
9)(33
3
x
y
x
y
x
y
y
x
Radicals and Their Properties
• Definition of nth Root of a Number.• Let a and b be real numbers and let n > 2 be a
positive integer. If
a = bn
then b is an nth root of a. If n = 2, the root is a square root. If n = 3, the root is a cube root.
Radicals and Their Properties
• Principal nth Root of a Number.• Let a be a real number that has at least one nth root.
The principal nth root of a is the nth root that has the same sign as a. It is denoted by a radical symbol
• The positive integer n is the index of the radical, and the number a is the radicand. If n = 2, omit the index and write .a
n a
Example 5
• Evaluate:
a) 36
Example 5
• Evaluate:
a)
636
36
Example 5
• Evaluate:
b) 36
Example 5
• Evaluate:
b)
636
36
Example 5
• Evaluate:
c) 36
Example 5
• Evaluate:
c)
DNE 36
36
Example 5
• Evaluate:
d) 3
64
125
Example 5
• Evaluate:
d) 3
64
125
4
5
64
125
64
1253
3
3
Example 5
• Evaluate:• You Try:
d) 3
8
27
Example 5
• Evaluate:• You Try:
d) 3
8
27
2
3
8
27
8
273
3
3
Example 5
• Evaluate:
e) 5 32
Example 5
• Evaluate:
e) 5 32
2325
Properties of Radicals
Properties of Radicals
Example 6
• Use the properties of radical to simplify each expression.
a) 28
Example 6
• Use the properties of radical to simplify each expression.
a) 28
41628
Example 6
• Use the properties of radical to simplify each expression.
b) 33 5
Example 6
• Use the properties of radical to simplify each expression.
b) 33 5
5555 133/133
Example 6
• Use the properties of radical to simplify each expression.
c) 3 3x
Example 6
• Use the properties of radical to simplify each expression.
c) 3 3x
xx 3/13
Example 6
• Use the properties of radical to simplify each expression.
d) 6 6y
||6 6 yy
Simplifying Radicals
• An expression involving radicals is in simplest form when the following conditions are satisfied.
1. All possible factors have been removed from the radical.
2. All fractions have radical-free denominators (accomplished by a process called rationalizing the denominator).
3. The index of the radical is reduced.
Example 7
• Simplify each radical.
a) 32
Example 7
• Simplify each radical.
a) 32
2421632
Example 7
• Simplify each radical.• You Try:
a) 24
Example 7
• Simplify each radical.• You Try:
a) 24
626424
Example 7
• Simplify each radical.
b) 4 48
Example 7
• Simplify each radical.
b) 4 48
4444 3231648
Example 7
• Simplify each radical.
c) 375x
Example 7
• Simplify each radical.
c) 375x
xxxxx 3532575 23
Example 7
• Simplify each radical.• You Try:
c) 548x
Example 7
• Simplify each radical.• You Try:
c) 548x
xxxxx 3431648 245
Example 8
• Simplify each radical.
a) 3 24
Example 8
• Simplify each radical.
a) 3 24
3333 323824
Example 8
• Simplify each radical.
b) 3 424a
Example 8
• Simplify each radical.
b) 3 424a
333 3333 4 323824 aaaaa
Example 8
• Simplify each radical.• You Try:
c) 3 640x
Example 8
• Simplify each radical.• You Try:
c) 3 640x
323 6333 6 525840 xxx
Example 9
• Combine each radical.
a) 273482
Example 9
• Combine each radical.
a) 273482
3933162273482
33938
Example 9
• Combine each radical.• You Try:
b) 3 43 5416 xx
Example 9
• Combine each radical.• You Try:
b) 3 43 5416 xx
33 3333 43 227285416 xxxxx
333 2)32(2322 xxxxx
Example 10
• Rationalize the denominator of each expression.
a) 3
5
Example 10
• Rationalize the denominator of each expression.
a) 3
5
3
35
3
3
3
5
You Try
• Rationalize the denominator of each expression.• You Try:
b) 2
1
Example 10
• Rationalize the denominator of each expression.• You Try:
b) 2
1
2
2
2
2
2
1
Example 11
• Rationalize the denominator of each expression.
a) 73
2
7379
)73(2
)73(
)73(
)73(
2
You Try
• Rationalize the denominator of each expression.• You Try:
b) 54
3
You Try
• Rationalize the denominator of each expression.• You Try:
b) 54
3
11
5312
)54(
)54(
)54(
3
Rational Exponents
• The numerator of a rational exponent denotes the power to which the base is raised, and the denominator denotes the index or the root to be taken.
Example 13
• Change the base from radical to exponential form.
a) 3
Example 13
• Change the base from radical to exponential form.
a) 3
2/133
Example 13
• Change the base from radical to exponential form.
b) 5x
Example 13
• Change the base from radical to exponential form.
b) 5x
2/52/155 )( xxx
You Try
• Change the base from radical to exponential form.• You Try:
c) 3 4y
You Try
• Change the base from radical to exponential form.• You Try:
c) 3 4y
3/43/143 4 )( yyy
Example 14
• Change the base from exponential to radical form.
a) 2/3)( yx
Example 14
• Change the base from exponential to radical form.
a) 2/3)( yx
32/3 )()( yxyx
You Try
• Change the base from exponential to radical form.
b) 4/14/3 yx
You Try
• Change the base from exponential to radical form.
b) 4/14/3 yx
4 34/134/14/3 )( yxyxyx
Example 15
• Simplify each rational expression.
a) 5/3)32(
Example 15
• Simplify each rational expression.
a) 5/3)32(
8)2())32(()32( 335/15/3
Example 15
• Simplify each rational expression.• You Try:
b) 3/2)27(
Example 15
• Simplify each rational expression.• You Try:
b) 3/2)27(
9
1)3())27(()27( 223/13/2
You Try
• Simplify each rational expression.• You Try:
c) 3/2)64(
You Try
• Simplify each rational expression.• You Try:
c) 3/2)64(
164))64(()64( 223/13/2
You Try
• Simplify each rational expression.• You Try:
d) 4/3)16(
You Try
• Simplify each rational expression.• You Try:
d) 4/3)16(
8
12))16(()16( 334/14/3