Download 2. - PMS-Math

8.1 Students will be able to evaluate powers that have zero and
negative exponents.
Evaluate the expression.
1
1. q3 when q =
4 ANSWER
2. c2 when c =
3
5
ANSWER
1
64
9
25
3. A magazine had a circulation of 9364 in 2001.
The circulation was about 125 times greater in 2006.
Use order of magnitude to estimate the circulation
in 2006.
ANSWER
about 106 or 1,000,000
8.1 Students will be able to evaluate powers that have zero and
negative exponents.
Homework Review
8.1 Students will be able to evaluate powers that have zero and
negative exponents.
Zero and Negative Exponents
8.1 Students will be able to evaluate powers that have zero and
negative exponents.
Notice what occurs when you divide powers
with the same base.
55555
55555
55
2
=
5

5
=
=
=
5
555
555
53
8.1 Students will be able to evaluate powers that have zero and
negative exponents.
a.
810 = 810 – 4
84
= 86
b.
(– 3)9 = (– 3)9 – 3
(– 3)3
= (– 3)6
c.
12
54 58 = 5
57
57
= 512 – 7
= 55
Use the quotient of powers property
8.1 Students will be able to evaluate powers that have zero and
negative exponents.
d.
x6
1
6
x = 4
x
x4
= x6 – 4
= x2
8.1 Students will be able to evaluate powers that have zero and
negative exponents.
Simplify the expression.
1.
611 11 – 5
=6
65
= 66
2.
(– 4)9
9–2
=
(–
4)
2
(– 4)
= (– 4)7
3.
94 93 = 97
92
92
= 97 – 2
= 95
8.1 Students will be able to evaluate powers that have zero and
negative exponents.
4.
y8
1
8
y = 5
y
y5
= y8 – 5
= y3
8.1 Students will be able to evaluate powers that have zero and
EXAMPLE
2
negative
exponents.
a.
b.
x 3 x3
y = y3
–7
x
2
=
–7
x
2
(– 7)2
49
=
=
x2
x2
8.1 Students will be able to evaluate powers that have zero and
negative exponents.
4x2
5y
a.
b.
5
2
a
b
3
(4x2)3
=
(5y)3
43 (x2)3
= 3 3
5y
64x6
=
125y3
(a2)5
1
2a2 = b5
a10
= 5
b
a10
=
2a2b5
8
= a
2b5
1
2a2
1
2a2
Power of a quotient property
Power of a product property
Power of a power property
Power of a quotient property
Power of a power property
Multiply fractions.
Quotient of powers property
8.1 Students will be able to evaluate powers that have zero and
negative exponents.
Simplify the expression.
5.
a 2 a2
= 2
b
b
5
6. = – y
7.
x2
4y
3
–5
=
y
2
3
2)2
(
x
=
(4y)2
x4
= 2 2
4y
x4
=
16y2
(– 5)3
125
= 3
=–
y
y3
Power of a quotient property
Power of a product property
Power of a power property
8.1 Students will be able to evaluate powers that have zero and
negative exponents.
8.
2s
3t
3
t5
16
23 s3
= 3 3
3 t
t5
16
8 s3 t 5
=
27 t3 16
8 s3 t5
=
27 16t3
s3 t 2
=
54
Power of a quotient property
Power of a power property
Multiply fractions.
8.1 Students will be able to evaluate powers that have zero and
negative exponents.
Fractal Tree
To construct what is known as a fractal tree, begin
with a single segment (the trunk) that is 1 unit long, as
in Step 0. Add three shorter segments that are 1 unit
2
long to form the first set of branches, as in Step 1.
Then continue adding sets of successively shorter
branches so that each new set of branches is half the
length of the previous set, as in Steps 2 and 3.
8.1 Students will be able to evaluate powers that have zero and
EXAMPLE
4
Solve a multi-step problem
negative
exponents.
a.
Make a table showing the number of new
branches at each step for Steps 1 - 4. Write the
number of new branches as a power of 3.
b.
How many times greater is the number of new
branches added at Step 5 than the number of new
branches added at Step 2?
8.1 Students will be able to evaluate powers that have zero and
EXAMPLE
4
Solve a multi-step problem
negative
exponents.
SOLUTION
a. Step Number of new branches
1
3 = 31
b.
2
9 = 32
3
27 = 33
4
81 = 34
The number of new branches added at Step 5 is
35. The number of new branches added at Step 2
is 32. So, the number of new branches added at
5
Step 5 is 3 = 33 = 27 times the number of new
32
branches added at Step 2.
8.1 Students will be able to evaluate powers that have zero and
for Example 4
GUIDED
PRACTICE
negative
exponents.
9. FRACTAL TREE In Example 4, add a column to the
table for the length of the new branches at each
step. Write the length of the new branches as
power of 1 . What is the length of a new branch
2
added at
Step 9?
SOLUTION
1
1 9
( 9 ) = 512 units
8.1 Students will be able to evaluate powers that have zero and
EXAMPLE
5
Solve a real-world problem
negative
exponents.
ASTRONOMY
The luminosity (in watts) of a
star is the total amount of
energy emitted from the star
per unit of time. The order of
magnitude of the luminosity
of the sun is 1026 watts. The
star Canopus is one of the
brightest stars in the sky. The
order of magnitude of the
luminosity of Canopus is 1030
watts. How many times more
luminous is Canopus than the
sun?
8.1 Students will be able to evaluate powers that have zero and
EXAMPLE
5
Solve a real-world problem
negative
exponents.
SOLUTION
Luminosity of Canopus (watts)
1030
30 - 26
4
10
10
=
=
=
Luminosity of the sun (watts)
1026
ANSWER
Canopus is about 104 times as luminous as the sun.
8.1 Students will be able to evaluate powers that have zero and
for Example 5
GUIDED
PRACTICE
negative
exponents.
9. WHAT IF? Sirius is considered the brightest star in the
sky. Sirius is less luminous than Canopus, but Sirius
1
appears to be brighter because it is much closer
to the
2
Earth. The order of magnitude of the luminosity of
Sirius is 1028 watts. How many times more luminous is
Canopus than Sirius?
SOLUTION
Luminosity of Canopus (watts)
1030
30 - 28
2
10
10
=
=
=
Luminosity of the sun (watts)
1028
ANSWER
Canopus is about 102 times as luminous as Sirius