Objectives

1.) To review and solidify basic exponential expressions and equations for the purpose of further use in more complex exponential problems

Vocabulary

• A power is a number resulting from a number brought to an exponent.

• The parts of a power: Include a base number and an exponent.

• The base is based, while the exponent floats

5 3 = 125

Warm- upSolve the following perfect square problems:

12 = 92 =22 = 102 =32 = 112 =42 = 122 =52 = 132 =62 =72 =82 =

Quick Study TimeYour skills on perfect squares, cubes, powers of 2 and

powers of 3 will be tested.Cubed Powers:13 =1 23 = 8 33 = 2743 = 64 53 = 125

Base 2 Powers21 = 2 22 = 4 23 = 824 = 16 25 = 32 26 = 64

Base 3 Powers31 = 3 32 = 9 33 = 2734 = 81

Definition of Exponential Equations

Exponential functions are equations involving constants with exponents

Notated: y = ax

a= base; a>0 and not equal to 1 x = exponent/ power

Properties of exponents

n

b

a

a n 1

an

anm anm

1.) a0 = 2.) aman =

3.) (ab)m = 4.) (an)m =

5.) 6.)

7.)

1 nma

mmba mna

n

n

b

a

In-depth Look of Property # 6Negative exponents

Cross the line, flip the sign.

In- depth Look of Property #7Radicals versus Rational Exponents

x12 x

x13 x3

x14 x4

x15 x5

x16 x6

...

Can you solve the expression

with your calculator?

40964

Putting it all together

a

b

n

an

bn

a n 1

an

anm anm

3.) (ab)m = amam 4.) (an)m = anm

1.) a0 = 1 2.) aman = am+n

1

x

4

3

Write the expression using positive rational exponent

5.) 6.)

7.)

Graphs of Exponential Functions

Pg. 200

Graphs of exponential functions

x

f(x)f(x) = ax , a>1

x

f(x)f(x) = ax , a<1f(x) = a-x

Characteristics of Exponential Function Graphs

Transformations

Compound interest

nt

n

rIP

1

One lucky day , you find $8,000 on the street. At the Bank of Baker- that’s my bank, I am offering you an interest rate of 10% a year. Being the smart students you are, you invest your money at my bank.

After the first year, your account collects 10% interest, so I would have to payout 8000+8000(.1)= $8,800

Or, 8,000(1 + .1) = $8,800

The second year, your $8,800 will collect even more interest and become

8,800(1 + .1) = 8,000(1 +.1)(1+.1)= $9,680

Complete the table below

Year 1 2 3 4 5Payout Amou

nt

8,800 9,680 10,648 11, 71212,884

One lucky day , you find $8,000 on the street. At the Bank of Baker- that’s my bank. I am offering you an interest rate of 10% a year. Being the smart students you are, you invest your money at my bank.

Deal or No Deal?

You come to me with $5000. I have an interest rate of 4.1 %. You want to establish this amount in my bank for 20 years.

What if I compound your investment quarterly. I will apply a compounded interest rate 4 times but I will divide the interest rate by 4.

trIP 1

20041.15000 P

20*4

4

041.15000

P

Initial investment

Interest rate in decimal form

I will pay 4 times per year for 20 years, but as consequence I will divide interest rate by 4

11,168.24

11,305.21

Compound interest

nt

n n

rI

1lim

In 1683, mathematician Jacob Bernoulli considered the value of

as n approaches infinity. His study was the first approximation of e

n

n

11

e= 2.718281828459045235460287471352662497757246093699959574077078727723076630353547594571382178525166427466391932003059921817413496629043572900338298807531952510190115728241879307…..

Comparable to an irrational number like ∏

\