1. 2.1 Using Properties of Exponents

2. 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
Negative Exponent Property: a-m= ; a≠0
Zero Exponent Property: a0=1; a≠0
Quotient of Powers: am = am-n; a≠ an
Power of Quotient: b≠0

3. Example 1 – Evaluate numerical expressions
Power of a product property
Power of a power property
Simplify and evaluate power. b.
115
118 –1
118
115 =
Negative exponent property
= 118 – 5
Quotient of powers property
= 113
= 1331
Simplify and evaluate power.

4. How many places did you have to move the decimal?
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?

5. Example – Scientific Notation
131,400,000,000 =
5,284,000
1.314 x 1011 =
5.284 x 106

6. 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.

7. The number of locusts is about 1.02 1011,
Write in scientific
notation.
Use multiplication
properties.
Product of powers
property
Write 10.2 in scientific
notation.
Product of powers
property
The number of locusts is about ,
or about 102,000,000,000.
ANSWER

8. Product of a powers property
You try…
2. (–8)(–8)3
SOLUTION
(–8)(–8)3
= (–8)(–8)3
Product of a powers property
= (–8)(–512)
Multiply
= 4096
Simplify

9. Power of a quotient property Simplify and evaluate power.
You try… 3.
2 3 9
SOLUTION
2 3 9 23 93 =
Power of a quotient property 8
729 =
Simplify and evaluate power.

10. You try… 6 • 10 – 4 4. 9 • 107 SOLUTION 6 •10 – 4 9 • 107
6 • 10 – 4
9 • 107 4.
SOLUTION
6 •10 – 4
9 • 107
quotient of power property
add power
Negative exponent property
Negative exponent property

11. Simplify expressions Product of powers property
a. b–4b6b7
= b–
= b9
Product of powers property b.
r–2 –3 s3
( r – 2 )–3
( s3 )–3 =
Power of a quotient property =
r 6
s–9
Power of a power property
= r6s9
Negative exponent property
c m4n –5
2n–5
= 8m4n – 5 – (–5)
Quotient of powers property
= 8m4n0= 8m4
Zero exponent property

12. 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 3
π (1500r)3
π r3
The volume of a
sphere is πr3.
Betelgeuse’s volume
Sun’s volume
4/3 = 4 3
π 15003r3
π r3
Power of a product
property

13. Zero exponent property
Quotient of powers
property =
Zero exponent property
= 3,375,000,000
Evaluate power.
Betelgeuse’s volume is about 3.4 billion times as great as the sun’s volume.
ANSWER

14. Simplify the expression. Tell which properties of exponents you used.
5. x–6x5 x3
SOLUTION
x–6x5x3
= x–6x5 + 3
Power of a product property
= x2
Simplify exponents.

15. Simplify the expression. Tell which properties of exponents you used.
(7y2z5)(y–4z–1)
SOLUTION
(7y2z5)(y–4z–1)
= (7y2z5)(y–4z–1)
Power of a product property
= (7y2 – 4)(z5 +(–1))
Simplify
= (7y–2)(z4)
Negative exponent property
= 7z4 y2

16. Simplify the expression. Tell which properties of exponents you used.7.
s 3 2
t–4
SOLUTION
s 3 2
t–4
(t–4 )2
s (3)2 =
Power of a product property
t–8 s6 =
Evaluate power.
s6t8 =
Negative exponent property

17. Simplify the expression. Tell which properties of exponents you used.8.
x4y–
x3y6
SOLUTION
x4y–
x3y6 =
(x4)3 (y–2)3
(x3)3(y6)3
Power of a powers property =
x12y–6
x9y18
Power of a powers property
Power of a Quotient property
= x3y–24 x3
y24 =
Negative exponent property

18. p. 91, 3-21 every third problem, 24-40 even
Assignment
p. 91, 3-21 every third problem, even