Transcript pps

Negative and Zero
Exponents
Lesson 5.2.1
1
Lesson
5.2.1
Negative and Zero Exponents
California Standards:
What it means for you:
Number Sense 2.1
Understand negative whole-number
exponents. Multiply and divide
expressions using exponents with
a common base.
You’ll learn what zero and
negative powers mean,
and simplify expressions
involving them.
Algebra and Functions 2.1
Interpret positive whole-number
powers as repeated multiplication and
negative whole-number powers as
repeated division or multiplication by
the multiplicative inverse. Simplify
and evaluate expressions that
include exponents.
Key words:
• base
• exponent
• power
2
Lesson
Negative and Zero Exponents
5.2.1
Up to now you’ve worked with only positive whole-number
exponents. These show the number of times a base is
multiplied. As you’ve seen, they follow certain rules and
patterns.
1212
83
3256
20
2–6
916
122
100
4–3
6–7
84
100
17–2
2–256
4–10
The effects of negative and zero exponents are trickier to
picture. But you can make sense of them because they
follow the same rules and patterns as positive exponents.
3
Lesson
5.2.1
Negative and Zero Exponents
Any Number Raised to the Power 0 is 1
Any number that has an exponent of 0 is equal to 1.
1
0
0
0
So, 2 = 1, 3 = 1, 10 = 1,
2
0
= 1.
For any number a  0, a0 = 1
You can show this using the division of powers rule.
4
Lesson
5.2.1
Negative and Zero Exponents
If you start with 1000, and keep dividing by 10,
you get this pattern:
1000 = 103
100 = 102
10 = 101
1 = 100
Now divide by 10: 103 ÷ 101 = 10(3 – 1) = 102
Now divide by 10: 102 ÷ 101 = 10(2 – 1) = 101
1 1÷÷10
1 1==10
(1(1
––
1)1)==10
00
Now divide
Now
divideby
by10:
10:10
10
10
10
10
The most important row is the second to last one.
When you divide 10 by 10, you have 101 ÷ 101 = 10(1 – 1) = 100.
You also know that 10 divided by 10 is 1.
So you can see that 100 = 1.
5
Lesson
5.2.1
Negative and Zero Exponents
This pattern works for any base.
For instance,
61 ÷ 61 = 6(1 – 1) = 60, and 6 divided by 6 is 1. So 60 = 1.
You can use the fact that any number to the power 0 is 1
to simplify expressions.
6
Lesson
5.2.1
Example
Negative and Zero Exponents
1
Simplify 34 × 30.
Leave your answer in base and exponent form.
Solution
34 × 30 = 34 × 1 = 34
You can use the multiplication of powers rule
to show this is right:
34 × 30 = 3(4 + 0) = 34
Add the exponents of the powers
You can see that being multiplied by 30 didn’t change 34.
7
Solution follows…
Lesson
5.2.1
Negative and Zero Exponents
Guided Practice
Evaluate the following.
1. 40
1
2. x0 (x  0)
1
3. 110 + 120
2
4. (7 + 6)0
1
5. 43 ÷ 43
1
6. y2 ÷ y2 (y  0)
1
7. 32 × 30
32 or 9
8. 24 × 20
24 or 16
9. a8 ÷ a0 (a  0)
a8
8
Solution follows…
Lesson
5.2.1
Negative and Zero Exponents
You Can Justify Negative Exponents in the Same Way
By continuing the pattern of powers shown below you can
begin to understand the meaning of negative exponents.
1000 = 103
100 = 102
10 = 101
1 = 100
Now divide by 10: 103 ÷ 101 = 10(3 – 1) = 102
Now divide by 10: 102 ÷ 101 = 10(2 – 1) = 101
Now divide by 10: 101 ÷ 101 = 10(1 – 1) = 100
9
Lesson
5.2.1
Negative and Zero Exponents
Carry on dividing each power of 10 by 10:
100 = 102
10 = 101
1 = 100
1
= 10–1
10
1
= 10–2
100
1
= 10–3
1000
Now divide by 10: 102 ÷ 101 = 10(2 – 1) = 101
Now divide by 10: 101 ÷ 101 = 10(1 – 1) = 100
Now divide by 10: 100 ÷ 101 = 10(0 – 1) = 10–1
Now divide by 10: 10–1 ÷ 101 = 10(–1 – 1) = 10–2
Now divide by 10: 10–2 ÷ 101 = 10(–2 – 1) = 10–3
10
Lesson
5.2.1
Negative and Zero Exponents
Look at the last rows, shown in red, to see the pattern:
1
= 10–1
10
1
= 10–2
100
1
= 10–3
1000
Now divide by 10: 10–1 ÷ 101 = 10(–1 – 1) = 10–2
Now divide by 10: 10–2 ÷ 101 = 10(–2 – 1) = 10–3
1
1
1–11
11
–2.–3.
One-tenth,
One-thousandth,
whichwhich
which
is ,iscan
is be
rewritten
, can
rewritten
as 1 =as10
as 2.=310
= 10
One-hundredth,
, can
bebe
rewritten
10 100
1000
10
1010
11
Lesson
Negative and Zero Exponents
5.2.1
This works with any number, not just with 10.
For example:
60 = 1
1
1
–1
=
and 1 ÷ 6 = , so 6 = .
6
6
1
1 1 1
1
–1
1
–2
–2
6 ÷ 6 = 6 and ÷ 6 = • = 2 = , so 6 = .
6
6 6 6
36
60 ÷
61
6–1
This pattern illustrates the general definition for
negative exponents.
1
–n
For any number a  0, a = n
a
12
Lesson
5.2.1
Example
Negative and Zero Exponents
2
Rewrite 5–3 without a negative exponent.
Solution
5–3
1
= 3
5
Using the definition of negative exponents
1
=
125
13
Solution follows…
Lesson
5.2.1
Example
Negative and Zero Exponents
3
1
Rewrite 5 using a negative exponent.
7
Solution
1
–5
=
7
75
Using the definition of negative exponents
14
Solution follows…
Lesson
Negative and Zero Exponents
5.2.1
Guided Practice
Rewrite each of the following without a negative exponent.
10. 7–3
1
73
11. 5–m
1
5m
12. x–2 (x  0)
1
x2
15
Solution follows…
Lesson
Negative and Zero Exponents
5.2.1
Guided Practice
Rewrite each of the following using a negative exponent.
1
13. 3 3–3
3
1
14. 4
6
6–4
1
15.
(q  0)
q×q×q
q–3
16
Solution follows…
Lesson
5.2.1
Negative and Zero Exponents
Independent Practice
Evaluate the expressions in Exercises 1–3.
1. 87020
1
2. g0 (g  0)
1
3. 20 – 30
0
17
Solution follows…
Lesson
Negative and Zero Exponents
5.2.1
Independent Practice
Write the expressions in Exercises 4–6 without
negative exponents.
1
45
4.
45–1
5.
x–6
(x  0)
6.
y–3
–
z–3
1
x6
(y  0, z  0)
1
1
–
y3 z3
18
Solution follows…
Lesson
5.2.1
Negative and Zero Exponents
Independent Practice
Write the expressions in Exercises 7–9 using
negative exponents.
1
8–2
7.
82
1
8. (r  0)
r6
r–6
1
9.
(p + q  0)
v
(p + q)
(p + q)–v
19
Solution follows…
Lesson
5.2.1
Negative and Zero Exponents
Independent Practice
In Exercises 10–12, simplify the expression given.
10. 54 × 50
54
11. c5 × c0 (c  0)
c5
12. f 3 ÷ f 0 (f  0)
f3
20
Solution follows…
Lesson
5.2.1
Negative and Zero Exponents
Independent Practice
13. The number of bacteria in a Petri dish doubles
every hour. The numbers of bacteria after each hour are
1, 2, 4, 8, 16, ... Rewrite these numbers as powers of 2.
20, 21, 22, 23, 24
1 1
1
14. Rewrite the numbers 1, , , and as powers of 2.
2 4
8
0
–1
–2
–3
2 , 2 , 2 , and 2
21
Solution follows…
Lesson
5.2.1
Negative and Zero Exponents
Round Up
So remember — any number (except 0) to the power of
0 is equal to 1. This is useful when you’re simplifying
expressions and equations.
Later in this Section, you’ll see how negative powers
are used in scientific notation for writing very small
numbers efficiently.
22