exponential growth laws of exponents and geometric patterns

Exponential Growth

Laws of Exponents and Geometric Patterns

Exponents

The ‘2’ is called the coefficient.The ‘x4’ is called the power,of which ‘x’ is the baseand ‘4’ is the exponent

Consider the expression

2x4 = 2(x)(x)(x)(x)

Laws of ExponentsMultiplying powerswith the same base:

Notice that:(a5)(a2) = (aaaaa)(aa)(a5)(a2) = aaaaaaa(a5)(a2) = a7

When multiplying powers with the same base, we

keep the base and add the exponents.

Laws of ExponentsDividing powerswith the same base:

Notice that:

43

7

3

7

3

7

bb

b

bbbbb

b

bbb

bbbbbbb

b

b

When dividing powers with the same base, we keep the base and subtract the

exponents.



Laws of Exponents

Power of a power:

When we have a power of a power, we keep the base and multiply the exponents.

Notice that:

623

23

223

)(

))(()(

)()(

ww

wwwwwww

wwww

When dividing powers with the same base, we keep the base and subtract the

exponents.



Using the Laws of Exponents

Simplify the following expressions using the first 3 exponent laws

a) (w3)(w10) b) (4d2)(10d6)c)

d) e) f)

6

8

3

15

s

s

2

)2)(2( 934

3

35

ts

ts 224

523

22

22

Answers:a) w13 b) 40d8 c) 5s2

d) 211 e) s8t8 f) 23

What does equal? 3

3

2

2

12

2

8

8

2

2

222

222

2

2

3

3

3

3

3

3

But we could also use an exponent law:

03

3

333

3

22

2

22

2

The value of any power with exponent 0 is 1.

12So 0

Zero and Negative ExponentsPowers of zero:

When dividing powers with the same base, we

keep the base and subtract the exponents.

Zero and Negative Exponents

What does equal? 4

2

3

3

2

23

2

3

2

4

2

3

1

3

1

3

3

33

1

3

3

3333

33

3

3

But we could also use an exponent law:

24

2

424

2

33

3

33

3

To change the sign of an exponent, change the base to

its reciprocal.(To change the base to its

reciprocal, change the sign of the exponent)

22

22

3

13

or3

13So

Negative powers:

When dividing powers with the same base, we

keep the base and subtract the exponents.

When multiplying powers with the same base, we keep the base and add the exponents.

When dividing powers with the same base, we keep the base and subtract the exponents.

The value of any power with exponent 0 is 1.

To change the sign of an exponent, change the base to its reciprocal.(To change the base to its reciprocal, change the sign of the exponent)

When we have a power of a power, we keep the base and multiply the exponents.

The 5Exponent Laws

bccb aa

cc

a

b

b

a

10 a

cbc

b

aa

a

cbcb aaa

Ex1.Simplify the following leaving no negative exponents (when possible).

a) 43

3

5

5 23

03

q

qb) ba

ba

x

x

3

2

c)

Answers: a) b) c)9

9

5

15 6q ba

ba

xx

22 1

or

Ex2. Evaluate (which means “get the value of the expression”)

a) b) c) d) e) 42

2

2

3

22 03

Answers: a) b) c) d) −1 e) 1 16

1

9

4

4

1


03


2

3

2a.

3

5

1b.

2

02

72

1c.

211 62d.

2

2

3

2

2

2

3

4

9

35125

22 12

214 25

2

5

1

25

1

2

61

21

2

61

63

2

64

2

32

223

49

Ex 3. Evaluate each expression

Changing the baseNotice that most of the exponent laws only work when there’s a common base. It is often helpful to be able to change the base of a power to match another power.

Ex. Simplify 2

3

16

84

xx

24

332

)2(

)2(2

xx

8

36

2

22

xx

8

9

2

2

x

892 x

All of these bases can be written as a base of 2.

This doesn’t look like simplifying…

…yet…

Yep, that’s simpler. It is now expressed as a single base.

This expression has some nasty values! You are NOT required to know 81-5 or (1/3)7. BUT…All of these bases can be written as a base of 3:

Ex 2. Evaluate without a calculator

74

52

31

9

8127

7142

5423

3)3(

33

1

2

4

3

33

1

39

381

327

78

206

3)3(

33

)3(

315

14

3

3

31

)15(14

Changing the base

An exponential equations is one where the variable (unknown) is in the exponent. The only technique we have so far to solve such equations is getting a common base. Soon we will learn another, stronger technique; logarithms.

Ex 1. Find the root(s), ie solve for x: 442 279 x

We must recognize that these bases can be rewritten using base 3.

43422 33 x

1284 33 x

1284 x

8124 x

5x

Solving Exponential Equations

Now, with two equal expressions, with equal bases, the exponents must also be equal.

Ex 2. Solve for x.

128

1864 43 xx

74336 222 xx

71218 222 xx

730 22 x

30

7

730

x

x


Recognize the common base of 2.

Now, with two equal expressions, with equal bases, the exponents must also be equal.

Ex 3. Find the root(s) of the equation below.

048)8(3 x

48)8(3 x

16)8( x

43 22 x

34

43

x

x


Isolate the power with the variable.

Recognize the common base of 2.

7172a. 22 x 256)2(4b. 3 x

42 84c. x

Your turn: Find the root(s) of the equations below.

0103d.2

xx


7172a. 22 x


Your turn: Solutions

2

42

622

22

642622

22

x

x

x

x

x

256)2(4b. 3 x

832 2)2(2 x

85 22 x

3

85

x

x

42 84c. x

0103d.2

xx

4322 22 x

1242 22 x

8

162

1242

x

x

x

0)2)(5( xx

0)5( x 0)2( x5x 2x


Your turn: Solutions

2or52

493

12

101433

2

4

2

2

xx

x

x

a

acbbx

Patterns (again)We have seen that patterns can be represented as equations:

Linear Pattern: dnttn )1(1

cbnantn 2 cbxaxy 2

bmxy

Quadratic Pattern:

common difference at level 1

common difference at level 2

But what about a pattern like this?

3, 6, 12, 24, 48

Geometric PatternsWe can quickly see that there is no common difference for this pattern……but there is a common ratio.

+3 +6 +12 +24

33 × 23 × 2 × 23 × 2 × 2 × 23 × 2 × 2 × 2 × 2

Or…33 × 21

3 × 22

3 × 23

3 × 24

Remember:20 = 1

× 20

×2 ×2 ×2 ×2

So we can express this pattern as:

3, 6, 12, 24, 48

+3 +6 +12

3, 6, 12, 24, 48Patterns with a common ratio are called geometric patterns.

Geometric PatternsWe see from this that this geometric pattern can be represented by the equation:

1)2(3 nnt

Let’s check: If n = 4

24

)8(3

)2(3

)2(3

4

4

34

144

t

t

t

t

3 × 20

3 × 21

3 × 22

3 × 23

3 × 24

The 2 is the common ratio of the pattern and is the base of the power in the equation.

The 3 is the first term of the pattern and is the coefficient in the equation.

3, 6, 12, 24, 48

Geometric PatternsIn general, a geometric pattern can be written using the equation

11 )( n

n rtt

The pattern 5, 10, 20, 40, 80,…can be represented by the equation 1

11

)2(5

)(

n

n

nn

t

rtt

We can check by plugging in n = 515

5 )2(5 t4

5 )2(5t)16(55 t

805 t

where t1 is the first term of the pattern (when n = 1) and where r is the common ratio.

1)2(5 nnt

Geometric Patterns PracticeFind the 10th term in each pattern:

a) 100, 50, 25, 12.5,…

b) 0.25, 1,4, 16,…

c) 5, 8, 11, 14, …

d) 1, −2, 4, −8, 16, …

Geometric Patterns Practice: Solutions

Find the 10th term in each pattern:

a) 100, 50, 25, 12.5,…

1

2

1100

n

nt

1953125.0

)001953125.0(100

2

1100

2

1100

10

10

9

10

110

10

t

t

t

t

a) CR = ½ t1 = 100



b) 0.25, 1, 4, 16,…

b) CR = 4 t1 = 0.25

1425.0 nnt

65536

)262144(25.0

425.0

425.0

10

10

910

11010

t

t

t

t



c) 5, 8, 11, 14, …

c) CD = 3 t1 = 5

3)1(5

)1(1

nt

dntt

n

n

32

)3)(9(5

3)110(5

10

10

10

t

t

t



d) 1, −2, 4, −8, 16, …

d) CR = −2 t1 = 1

121 nnt

512

2

2

10

910

11010

t

t

t

Geometric Patterns PracticeFor the pattern below, which term has a value of 768?

126 nnt

126768 n

12128 n

17 22 n

6, 12, 24, 48, …

17 n

8n The 8th term has a value of 768.

Speed of exponential growth:an old Indian legend

1 000 000

The last square requires more than 18,000,000,000,000,000,000 grains of rice, which is equal to about 210 billion tons and is allegedly sufficient to cover the whole territory of India with a meter thick layer of rice.At a production rate of ten grains of rice per square inch, the above amount requires rice fields covering twice the surface area of the Earth, oceans included.

http://www.singularitysymposium.com/exponential-growth.html

Exponential growth examplesA great many things in nature grow exponentially. Each of these situations can be modeled with a geometric pattern and thus an exponential equation.

http://www.youtube.com/watch?v=gEwzDydciWc

http://www.youtube.com/watch?v=gEwzDydciWc

Let’s set up a table to analyze the pattern.

Number of hours since the start

0 612

18

Number of bacteria present 1 2 4 8We see that the CR of this pattern is 2.

Ex. A certain type of bacteria doubles every six hours. The experiment begins with 1 bacteria.

However, this pattern involves n values that do not increase by 1; they increase by 6.

Also, the n values begin at 0, not 1….

When this happens, the equation must change….

Exponential growth examples

The old equation is for patternswhere n starts at 1, and increases by 1

11 )( n

n rtt

For most application problems(ie word problems),we’ll use this new equation:

periodx

rAy )(0

Where:x measures time since the starty is the amount at time x,r is the common ratio,A0 is the original amount (ie, the amount at time x = 0), and period is the amount by which the x values increase.

(Note, sometimes x is replaced by t to emphasize that it measures time.)


So we get thefunction

6)2(1x

y

For this question, we can see that: r = 2 (doubles)A0 = 1period = 6


Number of hours since the start

0 612

18

Number of bacteria present 1 2 4 8



6)2(1x

y

Now let x = 50 and solve for y.650

)2(1y

5398.322y

There would be 322 bacteria at time x = 50 hours.

a) How many bacteria are present after 50 hours?



b) When will there be 128 bacteria present?

Now let y = 128 and solve for x.

6)2(1x

y 6)2(1128x

6227x

67

x

42x At x = 42 hours there will be 128 bacteria.



c) When will there be 200 bacteria present?

Now let y = 200 and solve for x.6)2(1x

y 6)2(1200x

62200x

But 200 cannot be written as a power with base 2.We’ll need our stronger tool, logarithms, to solve this one.


exponential growth laws of exponents and geometric patterns

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