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11-EXT
11-EXTRational
RationalExponents
Exponents
Lesson Presentation
Holt
Algebra
Holt
Algebra
11
11-EXT
Rational Exponents
Objective
Simplify expressions containing
rational exponents.
Holt Algebra 1
11-EXT
Rational Exponents
Vocabulary
index
Holt Algebra 1
11-EXT
Rational Exponents
You have seen that taking
a square root and squaring
are the inverse operations
for nonnegative numbers.
There are inverse operations for other powers as well.
For example 3 represents a cube root, and it is the
inverse of cubing a number. To find 3 , look for three
equal factors whose product is 8. Since 2 • 2 • 2 = 8.
3
=2.
Holt Algebra 1
11-EXT
Rational Exponents
Operation
Inverse
Example
In the table, notice the small number to the left of the
radical sign. It indicate what root you are taking. This
is the index (plural, indices) of the radical. When a
radical is written without an index, the index is
understood to be 2. Only positive integers greater than
or equal to 2 may be indices of radicals.
Holt Algebra 1
11-EXT
Rational Exponents
Example 1: Simplifying Roots
Simplify each expression.
A.
B.
196
Think
3=
125
Think
4=
10,000
Think
6=
64
4
4
D.
2=
3
3
C.
Think
6
6
Holt Algebra 1
11-EXT
Rational Exponents
Check It Out! Example 1
Simplify each expression.
a. 3
3
b.
Think
3=
27
Think
5=
0
Think
4=
16
Think
2=
144
5
5
c.
4
4
d.
Holt Algebra 1
11-EXT
Rational Exponents
You have seen that exponents can be integers.
Exponents can also be fractions. What does it mean
when an exponent is a fraction? For example, what
is the meaning of
2
Product of Powers Property.
=3
So,
2
= 3. However,
= 3 also. Since
squaring either expression gives a result of 3, it
must be true that
Holt Algebra 1
.
11-EXT
Rational Exponents
Holt Algebra 1
11-EXT
Rational Exponents
Example 2: Simplifying
Simplify each expression
A.
3
=7
Use the definition of
Think
3=
.
343.
B.
5
=2
Holt Algebra 1
Use the definition of
Think
5=
32.
.
11-EXT
Rational Exponents
Check It Out! Example 2
Simplify each expression.
A.
2
= 11
Use the definition of
Think
2=
.
121.
B.
4
=3
Holt Algebra 1
Use the definition of
Think
4=
81.
.
11-EXT
Rational Exponents
Check It Out! Example 2
Simplify each expression.
C.
4
=4
Holt Algebra 1
Use the definition of
Think
4=
256
.
11-EXT
Rational Exponents
You can also have a fractional exponent with a
numerator other than one.
For example,
2
Product of Powers Property.
2
3
Definition of
2
= (2)
=4
Holt Algebra 1
Think
3=
8.
.
3
11-EXT
Rational Exponents
Example 3: Simplifying Expressions with Rational
Exponents
Simplify each expression.
A.
B.
5
4
= 243
Holt Algebra 1
5
2
Power of a Power
Property.
Definition of
.
5
=25
2
11-EXT
Rational Exponents
Example 3C: Simplifying Expressions with Rational
Exponents
Simplify the expression.
3
4
Product of Powers Property.
3
Definition of
Holt Algebra 1
.
11-EXT
Rational Exponents
Check It Out! Example 3
Simplify each expression.
a.
b.
Power of a Power
Property.
4
3
=(2)
=8
Holt Algebra 1
Definition of
.
5
=(1)2
=1
11-EXT
Rational Exponents
Check It Out! Example 3c
Simplify each expression.
Power of a Power
Property.
3
= (3) 4
= 81
Holt Algebra 1
Definition of
.