Transcript Math 9: Laws of Exponents

1. Products of Exponents
2. Quotients of Exponents
3. Negative Exponents
4. Evaluating Exponents
5. Scientific Notation
1. Products of Exponents
53 = 5∙5∙5
 So, 53∙ 54 = 5∙5∙5 ∙ 5∙5∙5∙5

53∙ 54 = 57
 We know that
 Do you see a relationship between
53∙ 54 and 57?
Exponent Law #1
For any real numbers a, m, and n:
a a  a
m
n
mn
Practice Problems
Now do practice problems 1 through 4 on your
handout.
We will go over them when you are done.
Now, go a step further…
what happens if we have
That would mean:
5  ?
7
5 5 5
7
7
3
7
From Exponent Law #1 we know that,
5
7 7 7
 5
21
Do you see a relationship between
5
7

3
and
5
21
?
Exponent Law #2
For any real numbers a, m, and n:
a 
m
n
a
mn
a
mn
When there is a product raised to a power,
raise each factor to the outside power.
Example:
7x 
3
5
 7 x
1 5
 7 x
5
3 5
15
Exponent Law #3
For any real numbers a, b, k, and m:
 ab 
k
m
 a b
m
km
Practice Problems
 Now do practice problems 5 through 8 on your
handout.
 We will go over them when you are done.
1. Products of Exponents
2. Quotients of Exponents
3. Negative Exponents
4. Evaluating Exponents
5. Scientific Notation
2. Dividing with exponents
5
5  5  5 5555
5
4


 5
3
5
1
5  5  5 1
7
4
Do you see a relationship between
7
5
3
5
and
5
4
?
Exponent Law #4
For any real numbers a, k, and m,
where a ≠ 0:
k
k m
m
a
a

a
Note: the same base number in the
numerator and denominator
Exponent Law #4 Corollary
Any real number raised to the zero power is 1.
3
5
125

 1
3
5
125
and by Law #4
3
5
3 3
0
 5
 5
3
5
So,
5 1
0
Exponent Law #5
For any real numbers a, b, k, m and w:
m
a
a

m
b
b
 
m
a 
a

 bw 
wm
b
 
k
and
m
km
Practice Problems
 Now do practice problems 9 through 11 on your
handout.
 We will go over them when you are done.
1. Products of Exponents
2. Quotients of Exponents
3. Negative Exponents
4. Evaluating Exponents
5. Scientific Notation
3. Negative Exponents
2
5
Consider 56
and use Law #4.
2
5
2 6
4
 5
 5
6
5
But what does
5
4
mean?
5
5  5 1
1

 4
6
5
5  5 5555 5
2
1
Thus, 5  4
5
4
Which leads us to the next exponent
law…
Exponent Law #6
For any real numbers x, m, and k, where x
≠ 0:
x
m
1
 m
x
and
1
k

x
k
x
However, this law comes with a warning….
CAUTION!!
WARNING!!
It is very easy to make mistakes!
3
What is wrong with…
a
4
a
7
a
?
CAUTION!!
WARNING!!
Be careful with the minus sign…
3
a
1
3  7
4
a
 a or 4
7
a
a
Not a
4
CAUTION!!
WARNING!!
Another easy sign error…
5
What’s wrong with….
a
3
a
2
a
?
CAUTION!!
WARNING!!
Be careful with the negative signs…
5
a
5  2 
52
7
a
a a
2
a
Not a
3
Practice Problems
 Now do practice problems 12 – 14 on your
handout.
 We will go over them when you are
finished.
More Practice Problems
 Now you can do practice problems 15
through 18 on your handout.
 These problems will use all the Exponent
Laws we have learned so far.
 We will go over them when you are done.
1. Products of Exponents
2. Quotients of Exponents
3. Negative Exponents
4. Evaluating Exponents
5. Scientific Notation
Consider this equation:
Reduce the fraction:
Divide both sides by 2:
2x 5
 1458
2
x
2  x  1458
3
2 x  1458
3
x  729
5 2
Now, what times itself 3 times equals 729?
93  729
x 9
3
3
Thus, x = 9
Practice Problems
 Now do practice problems 19 through 21
on your handout.
 We will go over them when you are done.
1. Products of Exponents
2. Quotients of Exponents
3. Negative Exponents
4. Evaluating Exponents
5. Scientific Notation
5. Scientific Notation
 Used for very large & very small numbers.
 Makes multiplying & dividing much easier.
m
_.
_
_
_
x
10
 Has this form:
 Has 1 non-zero digit left of the decimal point.
 Move the decimal point counting the moves.
 Moving left means a positive exponent.
 Moving right means a negative exponent.
Write 8,532,000 in scientific notation.
1. Locate the decimal point.
2. Move the decimal point.
8,532,000.
8.532
3. Count number of places moved and direction.
6 places to the left.
4. Make the number of places moved the
exponent.
5. Write the number.
10
6
8,532,000  8.532 x 106
Write 0.0000345 in scientific notation.
1. Locate the decimal point.
2. Move the decimal point.
3.45
3. Count the number of places moved and
direction.
5 places to the right.
4. Make the number of places moved the
10
5
exponent.
5. Write the number. 0.0000345  3.45 x 10 5
Multiplying and Dividing with scientific
notation.
 Consider :
 4.3 x 10  2.1 x 10 
3
15
 Separate the decimals and powers of 10.
 4.3  2.1 x 10 10 
15
3
 Multiply decimals and powers of 10 separately.
15  3 
9.03 x 10
9.03 x 1012
Another example:
 Consider .0000000000063 .042
 Put numbers into scientific notation.
 6.3 x 10  4.2 x 10 
12
2
 Separate decimals and powers of 10. Then
multiply.
12  2 
 6.3  4.2 
x 10
26.46 x 10 14
WARNING!! THIS IS NOT SCIENTIFIC NOTATION!
Put 26.46 x 10
14
into scientific notation.
 Move the decimal counting places moved
and direction.
2.646 x 10
14 1
2.646 x 10
13
 Thus,
.0000000000063.042  2.646 x 10
13
Practice Problems
 Now do practice problems 22 through 27
on your handout.
 We will go over them when you are done.
Review Lessons Worksheet
 You are now ready to do the Worksheet for
these lessons.
 It is worth 20 points toward passing Math 9.
 When is it due???