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

Homework

• Pages 21-22

• 5-11 odd

• 17-35 odd

• 51-72 multiples of 3