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2.1 Using
Properties of
Exponents
p. 89
Properties of Exponents
a&b are real numbers, m&n are integers
•
Product Property: am * an=am+n
•
Power of a Power Property: (am)n=amn
•
Power of a Product Property: (ab)m=ambm
1
a-m= a m
•
Negative Exponent Property:
•
•
Zero Exponent Property: a0=1; a≠0
Quotient of Powers: am = am-n; a≠0
an
Power of Quotient:  a  m a m b≠0
•
   m
b
b
; a≠0
Example 1 – Evaluate numerical
expressions
Power of a product property
Power of a power property
Simplify and evaluate power.
b.
115 –1 118
=
118
115
Negative exponent property
= 118 – 5
= 113
= 1331
Quotient of powers property
Simplify and evaluate power.
Scientific Notation
• 131,400,000,000=
1.314 x 1011
Put that number here!
Move the decimal
behind the 1st number
How many
places did you
have to move
the decimal?
Example – Scientific Notation
• 131,400,000,000 =
•
5,284,000
1.314 x 1011 =
5.284 x 106
1.314
5
11 6
*10
 .249 *10  24,900
5.284
Use scientific notation in
real life
A swarm of locusts may contain as
many as 85 million locusts per square
kilometer and cover an area of 1200
square kilometers. About how many
locusts are in such a swarm?
SOLUTION
Substitute values.
Write in scientific
notation.
Use multiplication
properties.
Product of powers
property
Write 10.2 in scientific
notation.
Product of powers
property
ANSWER
The number of locusts is about 1.02
or about 102,000,000,000.
1011,
You try…
2.
(–8)(–8)3
SOLUTION
(–8)(–8)3 = (–8)(–8)3
Product of a powers property
= (–8)(–512)
Multiply
= 4096
Simplify
You try…
3.
2
9
3
SOLUTION
2
9
3
3
2
= 3
9
=
8
729
Power of a quotient
property
Simplify and evaluate power.
You try…
4.
6 • 10 – 4
9 • 107
SOLUTION
6 •10 – 4 6
9 • 107 = 9
=
• 10 – 4 – 7 quotient of power
property
6
=
9
• 10 – 11
2
=
3
• 10 – 11Negative exponent property
2
3 1011
add power
Negative exponent property
Simplify expressions
a.
b.
b–4b6b7
r–2
= b–4 + 6 + 7
–3
s3
=
= b9
( r – 2 )–3
( s3 )–3
r6
=
s–9
= r6s9
c.
16m4n –5
2n–5
= 8m4n – 5 – (–5)
= 8m4n0= 8m4
Product of powers property
Power of a quotient property
Power of a power property
Negative exponent property
Quotient of powers property
Zero exponent property
Astronomy
Betelgeuse is one of the stars
found in the constellation
Orion. Its radius is about 1500
times the radius of the sun.
How many times as great as the
sun’s volume is Betelgeuse’s
volume?
Let r represent the sun’s radius. Then 1500r represents Betelgeuse’s
radius.
4
The volume of a
π (1500r)3
Betelgeuse’s volume
3
4/3πr3.
sphere
is
=
Sun’s volume
4
π r3
4 3
π 15003r3
3
Power of a product
= 4
property
π r3
3
= 15003r0
= 15003
Quotient of powers
property
1
Zero exponent property
= 3,375,000,000 Evaluate power.
ANSWER
Betelgeuse’s volume is about 3.4 billion
times as great as the sun’s volume.
Simplify the expression. Tell which
properties of exponents you used.
5.
x–6x5 x3
SOLUTION
x–6x5x3
= x–6x5 + 3
= x2
Power of a product property
Simplify exponents.
Simplify the expression. Tell which
properties of exponents you used.
6.
(7y2z5)(y–4z–1)
SOLUTION
(7y2z5)(y–4z–1)
= (7y2z5)(y–4z–1)
= (7y2 – 4)(z5 +(–1))
= (7y–2)(z4)
=
7z4
y2
Power of a product property
Simplify
Negative exponent property
Simplify the expression. Tell which
properties of exponents you used.
7.
s3
2
t–4
SOLUTION
s3
t–4
2
=
=
=
s (3)2
(t–4 )2
s6
t–8
s6t8
Power of a product property
Evaluate power.
Negative exponent property
Simplify the expression. Tell which
properties of exponents you used.
8.
x4y–2
3
x3y6
SOLUTION
x4y–2
x3y6
3
=
=
(x4)3 (y–2)3
(x3)3(y6)3
x12y–6
Power of a powers property
x9y18
= x3y–24
=
Power of a powers property
x3
y24
Power of a Quotient property
Negative exponent property
Assignment
p. 91, 3-21 every third
problem, 24-40 even