exponential growth laws of exponents and geometric patterns
TRANSCRIPT
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.
When multiplying powers with the same base, we
keep the base and add 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.
When multiplying powers with the same base, we
keep the base and add 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
Zero and Negative Exponents
03
Zero and Negative Exponents
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
Solving Exponential Equations
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
Solving Exponential Equations
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
Solving Exponential Equations
7172a. 22 x
Solving Exponential Equations
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
Solving Exponential Equations
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
Geometric Patterns Practice: Solutions
Find the 10th term in each pattern:
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
Geometric Patterns Practice: Solutions
Find the 10th term in each pattern:
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
Geometric Patterns Practice: Solutions
Find the 10th term in each pattern:
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
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.)
Exponential growth examples
So we get thefunction
6)2(1x
y
For this question, we can see that: r = 2 (doubles)A0 = 1period = 6
Exponential growth examples
Number of hours since the start
0 612
18
Number of bacteria present 1 2 4 8
Ex. A certain type of bacteria doubles every six hours. The experiment begins with 1 bacteria.
Ex. A certain type of bacteria doubles every six hours. The experiment begins with 1 bacteria.
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?
Exponential growth examples
Ex. A certain type of bacteria doubles every six hours. The experiment begins with 1 bacteria.
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.
Exponential growth examples
Ex. A certain type of bacteria doubles every six hours. The experiment begins with 1 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 examples