exponents powers and exponents. 9/13/2013 exponents 2 integer powers for any real number a, how can...

Exponents

Powers and

Exponents

Exponents 29/13/2013

Integer Powers For any real number a , how can we represent a • a or a • a • a ? By convention, we write a • a = a2 and a • a • a = a3 In general, for any positive integer n ,

the nth power of a is a • a • … • a

Powers

n factors

= a n


Integer Powers Examples:

3 • 3 = 32 = 9 (–3) • (–3) = (–3)2 = 9 2 • 2 • 2 • 2 • 2 = 25 = 32

Powers


The nth Power For any real number a , and positive integer n we write the nth power of a as

Powers and Exponents

a n

Base a Exponent n

Base 3 Exponent 4

3 4

E

Example: 4th power of 3


Combining Exponents Consider the following:

32 • 33

Exponents

= (3 • 3) • (3 • 3 • 3) = 3 • 3 • 3 • 3 • 3

5 factors

of 3

= 35 2

factors

of 3

3 factors

of 3

Notice that 5 = 2 + 3

Is this significant ?


Combining Exponents Let’s try 3 • 33

3 • 3 3

Exponents

= (3) • (3 • 3 • 3)

3 factors

of 3

1 factor

of 3

= 3 • 3 • 3 • 3

4 factors

of 3

= 3 4

Question:For positive integers m and n consider

3m • 3n = 3r What can be said about r ?


Addition of Exponents: General Rule To multiply powers of the same base, add the exponents:

a m • a

n = a m + n

For what bases and exponents does this work?

As demonstrated, this works for positive base and positive integer exponents

Exponents


Addition of Exponents: General Rule

a m • a

n = a m + n

Exponents

16 = 24 8 = 23 4 = 22 2 = 21 1 = 1/2 = 2-1

1/4 = 2-2 1/8 = 2-3

OR

81 = 34 27 = 33 9 = 32 3 = 31 1 = 1/3 = 3-1 1/9 = 3-2 1/27 = 3-3

OR ...20 30

C Consider:

? ?


Negative Exponents: General Rule We have seen that

Exponents

21

1=2-1 2

1=

a –m

a m

1=

22

1=2-2 4

1=

23

1=2-3 8

1=

General Rule:

31

1=3-1 3

1=

32

1=3-2 9

1=

33

1=3-3 27

1=

OR


Addition of Negative Exponents What is 3-2 • 3-3 ?

Consider

Exponents

32+3

1=35

1= 3-5 =

33

1•

32

1=3-2 3-3•33•32

1=

for m, n both positive OR both negative

a m • a

n = a m + n GGeneral Rule:


Positive and Negative Exponents What is 32 • 3-3 ?

Exponents

31

1= 3–1 =

32+1

32

=

= 32

•32 31

3 =

33

1•32=32 3-3 • CConsider:

33

32

=

33 3-2 •

32

1•33=32

33

=

32

32+1

= =32

•32 31

31 =


Positive and Negative Exponents What is 32 • 3-3 ?

Exponents

33 3-2 • 31 =

GGeneral Rule: for any integers m, n

a m

• a –n = a

m – n


The Zero Exponent We saw that 20 = 1 and 30 = 1 Is this true in general for any a0 ?

For any non-zero a and any integer m

Exponents

am

am= = 1

Undefined !!

a0 = am – m

Then what is 00 ? WIf a = 0 ?

a0 = 1


Division of Powers

For non-zero base a and any integers

m and n, what is

Power Functions

am

an = am

an

1• = am a–n• = am – n

am

an ?

RRewriting:


Division of Powers

Power Functions

am

an = am

an

1• = am a–n• = am – n

am

an = am – n

F For non-zero base a and integers m , n

GGeneral Division Rule:


Examples: Rewrite the following in simplified form:

1.

Power Functions

x7

x4

x • x • x • x • x • x • xx • x • x • x=

(x • x • x • x)(x • x • x • x)= • (x • x • x)

1= • (x • x • x)

= x3

(x • x • x • x) • (x • x • x)(x • x • x • x)=


Examples: Rewrite the following in simplified form:

2.

3.

Power Functions

x7

x4 = x7 – 4

=x3

x4

x7 = x4 – 7

= x–3


Powers of Powers Example:

82 = (23)2

= 23 • 23

= (2 • 2 • 2) • (2 • 2 • 2)

= 2 • 2 • 2 • 2 • 2 • 2

= 26

= 23 •2

Power Functions


Powers of Powers In General:

Suppose we have bn and told that b = am

How do we find bn as a power of a ?

This is just a power of a power:

bn = (am)n

= am • n = amn

Power Functions


Mixed Bases and Powers Examples:

1. (a • b)3 = (a • b) • (a • b) • (a • b)

= (a • a • a) • (b • b • b)

= a3 • b3

= a3b3

Power Functions



2. (x2y4)3 = (x2)3(y4)3

= x6y12

Does this work if, from 1. , we have

a = x2 and b = y4 ?

Power Functions

Question:


Mixed Bases and Powers Examples: 3.

Power Functions

x2

y2=

= •xy

xy

=•x x

y y•

2( xy)



4.

5.

Power Functions

=•

x3

y5

x3

y5

• =x6

y10

=a8x6

b–4y10

= a8b4x6

y10

2

( x3

y5) = •x3

y5

x3

y5

2

( a4x3

b–2y5)


Mixed Bases and Powers General Rules:

1. For any real numbers a and b and any integer n

(a • b)n = an • bn

2. For any real numbers a and b, with b non-zero, and any integer n

Power Functions

=bnan

( ab

n

)


Power Functions Definition:

A power function f(x) is defined by

f(x) = xb

for some constant b Examples:

f(x) = x2

f(x) = x–4

f(x) = x2/3

Power Functions


Radical Functions Definition:

A radical function f(x) is defined by

f(x) = xb

for b = 1/n for some integer n ≥ 2

Examples:f(x) = x1/2

f(x) = x1/3

Power Functions

x=

x3=


Rational Exponent EquationsSolve:

1. x1/4 = 3

2. 2x1/3 – 5 = 1

3. n–2 + 3n–1 + 2 = 0

Power Functions


Think about it !

exponents powers and exponents. 9/13/2013 exponents 2 integer powers for any real number a, how can...

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