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2.1 Properties of Exponents
Warm Up
Lesson Presentation
Lesson Quiz
Holt Algebra 2
Warm Up
Simplify.
1. 4  4  4
3.
64
20
5.
7. 3  104
2.
4.
6. 105
30,000
100,000
Objectives
Simplify expressions involving
exponents.
Use scientific notation.
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.
When the base includes more than one
symbol, it is written in parentheses.
Reading Math
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.”
Check It Out! Example 1a
Write the expression in expanded form.
(2a)5
(2a)5
(2a)(2a)(2a)(2a)(2a)
The base is 2a, and
the exponent is 5.
2a is a factor 5 times.
Check It Out! Example 1b
Write the expression in expanded form.
3b 4
3b4
The base is b, and the exponent is
4.
3  b  b  b  b b is a factor 4 times.
Check It Out! Example 1c
Write the expression in expanded form.
–(2x – 1)3y
2
–(2x – 1)3y
2
There are two bases: 2x–1,
and y.
–(2x – 1)(2x – 1)(2x – 1)  y  y
2x–1 is a factor 3
times, and y is a
factor 2 times.
Caution!
Do not confuse a negative exponent with a negative expression.
Check It Out! Example 2a
Simplify the expression.
32
33=9
The reciprocal of
.
Check It Out! Example 2b
Write the expression in expanded form.
(–5)–5
 1  5
 5 

The reciprocal of
.
Check It Out! Example 3a
Simplify the expression. Assume all variables
are nonzero.
(5x 6)3
53(x 6)3
Power of a Product
125x
Power of a Power
(6)(3)
125x18
Check It Out! Example 3b
Simplify the expression. Assume all variables
are nonzero.
(–2a3b)–3
Negative Exponent Property
Power of a Power
Simplify expressions
EXAMPLE
a.
b.
b
–4b 6b 7=
r –2
s3
b
–4 + 6 + 7
16m 4n
2n –5
9
Product of powers property
–3
= ( r – 2 )–3
( s3 )–3
Power of a quotient property
6
r
=
s –9
Power of a power property
= r 6s
c.
=b
–5
9
= 8m 4n
Negative exponent property
– 5 – (–5)
= 8m 4n 0= 8m
Quotient of powers property
4
Zero exponent property
Standardized Test Practice
EXAMPLE 4
SOLUTION
(x
–3y 3)2
x 5y
6
=
(x
–3)2(y 3)2
x
5y 6
–6y 6
x
=
x 5y 6
Power of a product property
Power of a power property
Standardized Test Practice
EXAMPLE
=x
– 6 – 5y 6 – 6
Quotient of powers property
=x
–11y 0
Simplify exponents.
=x
–11
= 1 11
x
ANSWER
1
Zero exponent property
Negative exponent property
The correct answer is B.
GUIDED PRACTICE
Simplify the expression. Tell which properties of
exponents you used.
x
–6x 5
x
3
SOLUTION
x
–6x 5x 3=
x –6x 5 + 3
=x
2
Power of a product property
Simplify exponents.
GUIDED PRACTICE
(7y 2z 5)(y
–4z –1)
SOLUTION
(7y 2z 5)(y
–4z –1)=
(7y 2z 5)(y
–4z –1)
Power of a product property
= (7y
2 – 4)(z 5 +(–1))
Simplify
= (7y
–2)(z 4)
Negative exponent property
=
7z
y
2
4
GUIDED PRACTICE
s3
t -4
2
SOLUTION
s3
t –4
2
=
s (3)2
t (-4)2
Power of a product property
=
s6
t –8
Evaluate power.
=
s 6t
8
Negative exponent property
GUIDED PRACTICE
x 4y –2 3
x 3y 6
SOLUTION
x 4y –2 3 (x 4)3 (y –2)3
=
3
6
x y
(x 3)3(y 6)3
x 12y –6
=
x 9y 18
= x 3y
=
y
x3
24
–24
Power of a powers property
Power of a powers property
Power of a Quotient property
Negative exponent property
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.
Check It Out! Example 4a
Simplify the expression. Write the answer in scientific
notation.
0.25  10–3
2.5 
10–4
Divide 2.325 by 9.3 and subtract
exponents: 6 – 9 = –3.
Because 0.25 < 10, move the decimal
point right 1 place and subtract 1 from
the exponent.
Check It Out! Example 4b
Simplify the expression. Write the answer in
scientific notation.
(4  10–6)(3.1  10–4)
(4)(3.1)  (10–6)(10–4)
12.4  10–10
1.24 
10–9
Multiply 4 by 3.1 and add
exponents: –6 – 4 = –10.
Because 12.4 >10, move the
decimal point left 1 place and
add 1 to the exponent.
EXAMPLE
Use scientific notation in real life
Locusts
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
= 85,000,000
1200
Substitute values.
EXAMPLE
Use scientific notation in real life
= (8.5 107)(1.2
103) Write in scientific
notation.
= (8.5 1.2)(107 103) Use multiplication
properties.
= 10.2
= 1.02
1010
101
Product of powers
property
1010 Write 10.2 in scientific
notation.
= 1.02
ANSWER
1011
Product of powers
property
The number of locusts is about 1.02
or about 102,000,000,000.
1011,
Lesson Quiz
1. Multiply (5.4
104)(2.5 10–7). Write the
answer in scientific notation.
1.35
ANSWER
10–2
Simplify the expression. Tell which properties of
exponents you used.
2b –5
a
b
2.
3a 5 a –4
ANSWER
a
3b
4
; Power of a product, power of a
power, quotient of powers, negative
exponent properties
Daily Homework Quiz
3. The mass of Saturn is about 5.7
1026
kilograms.
The mass of Jupiter is about 3.7 1027
kilograms.
About how many times greater is Jupiter’s
mass?
ANSWER
about 6.5 times