1. Use Properties of Exponents Lesson 2.1
Honors Algebra 2
Use Properties of Exponents
Lesson 2.1

2. Goals Goal Rubric Simplify expressions involving powers.
Use scientific notation.
Level 1 – Know the goals.
Level 2 – Fully understand the goals.
Level 3 – Use the goals to solve simple problems.
Level 4 – Use the goals to solve more advanced problems.
Level 5 – Adapts and applies the goals to different and more complex problems.

3. Vocabulary
Scientific Notation

4. Powers & Exponents
In an expression of the form an, a is the base, n is the exponent, and the quantity an is called a power. The exponent indicates the number of times that the base is used as a factor.

5. Powers & Exponents
When the base includes more than one symbol, it is written in parentheses.
A power includes a base and an exponent. The expression 23 is a power of 2. It is read “2 to the third power” or “2 cubed.”
Reading Math

6. Properties of Exponents

7. Properties of Exponents

8. Simplifying Expressions
The properties of exponents are used to simplify algebraic expressions.
A simplified expression contains only positive exponents.

9. EXAMPLE 1
Evaluate numerical expressions a.
(– )2
= (– 4)2 (25)2
Power of a product property =
Power of a power property =
= 16,384
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.

10. Evaluate the expression.
Your Turn:
for Examples 1
Evaluate the expression.
(42 )3
2. (–8)(–8)3
4096
ANSWER
Product of a powers property
Power of a power property

11. Evaluate the expression.
Your Turn:
for Examples 1
Evaluate the expression. 3.
2 3 9 8
729
Power of a quotient property
ANSWER
– 4 4.
quotient of power property 2
ANSWER
Negative exponent property

12. a. b–4b6b7 = b–4 + 6 + 7 = b9 r–2 –3 s3 ( r – 2 )–3 ( s3 )–3 = = r 6
EXAMPLE 2
Simplify expressions
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

13. (x–3y3)2 (x–3)2(y3)2 x5y6 x5y6 x –6y6 x5y6 Power of a product property
EXAMPLE 3
Standardized Test Practice
SOLUTION
(x–3y3)2
x5y6 =
(x–3)2(y3)2
x5y6
Power of a product property
x –6y6
x5y6 =
Power of a power property

14. = x – 6 – 5y6 – 6 = x–11y 0 = x–11 1 x11 The correct answer is B.
EXAMPLE 3
Standardized Test Practice
= x – 6 – 5y6 – 6
Quotient of powers property
= x–11y 0
Simplify exponents.
= x–11 1
Zero exponent property = 1
x11
Negative exponent property
The correct answer is B.
ANSWER

15. EXAMPLE 4
Compare real-life volumes
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?
Astronomy

16. EXAMPLE 4
Compare real-life volumes
SOLUTION
Let r represent the sun’s radius. Then 1500r represents Betelgeuse’s radius. = 4 3
π (1500r)3
π r3
Betelgeuse’s volume
Sun’s volume
The volume of a
sphere is πr3. 4 3 = 4 3
π 15003r3
π r3
Power of a product
property

17. EXAMPLE 4
Compare real-life volumes
= 15003r0
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

18. Simplify the expression.
Your Turn:
for Examples 2, 3, and 4
Simplify the expression.
5. x–6x5 x3
Product of powers property x2
ANSWER
6. (7y2z5)(y–4z–1)
7z4 y2
Product of powers property
Negative exponent property
ANSWER

19. Simplify the expression.
Your Turn:
for Examples 2, 3, and 4
Simplify the expression. 7.
s 3 2
t–4
s6t8
Power of a quotient property
Power of a power property
Negative exponent property
ANSWER 8.
x4y–
x3y6
Quotient of powers property
Power of a product property
Power of a power property x3
y24
Negative exponent property
ANSWER

20. Scientific Notation
Scientific notation is a method of writing numbers by using powers of 10. In scientific notation, a number takes a form m  10n, where 1 ≤ m <10 and n is an integer.
You can use the properties of exponents to calculate with numbers expressed in scientific notation.

21. EXAMPLE 5
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?
Locusts
SOLUTION
= 85,000,
Substitute values.

22. The number of locusts is about 1.02 1011, or about 102,000,000,000.
EXAMPLE 5
Use scientific notation in real life
= ( )( )
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

23. Simplify the expression. Write the answer in scientific notation.
Example 6
Simplify the expression. Write the answer in scientific notation.
Divide 4.5 by 1.5 and subtract exponents: –5 – 6 = –11.
3.0  10–11

24. Simplify the expression. Write the answer in scientific notation.
Your Turn:
Simplify the expression. Write the answer in scientific notation.
Solution
Divide by 9.3 and subtract exponents: 6 – 9 = –3.
0.25  10–3
2.5× 10 −1 × 10 −3
Because 0.25 < 10, move the decimal point right 1 place and subtract 1 from the exponent.
2.5  10–4

25. Simplify the expression. Write the answer in scientific notation.
Your Turn:
Simplify the expression. Write the answer in scientific notation.
(4  10–6)(3.1  10–4)
Solution
(4)(3.1)  (10–6)(10–4)
Multiply 4 by 3.1 and add exponents: –6 – 4 = –10.
12.4  10–10
1.24× 10 1 × 10 −10
Because 12.4 >10, move the decimal point left 1 place and add 1 to the exponent.
1.24  10–9