Download Rational Exponents

Roots, Radical Expressions,
and Radical Equations
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8
Section
8.7
Rational Exponents
Copyright © Cengage Learning. All rights reserved.
Objectives
1 Simplify a numerical expression containing
a rational exponent with a numerator of 1.
2 Simplify a numerical expression containing
a rational exponent with a numerator other
than 1.
3 Apply the rules of exponents to an expression
containing rational exponents.
3
1. Simplify a numerical expression containing
a rational exponent with a numerator of 1
4
Simplify a numerical expression containing a rational exponent
with a numerator of 1
We have seen that a positive integer exponent indicates
the number of times that a base is to be used as a factor in
a product.
For example, x4 means that x is to be used as a factor four
times.
4 factors of x
x4 = x  x  x  x
5
Simplify a numerical expression containing a rational exponent
with a numerator of 1
Furthermore, we recall the following rules of exponents.
If m and n are natural numbers and x, y  0, then
xmxn = xm +n
(xm)n = xmn
(xy)n = xnyn
x0 = 1
6
Simplify a numerical expression containing a rational exponent
with a numerator of 1
To give meaning to rational (fractional) exponents, we
consider
. Because
is the positive number whose
square is 7, we have
We now consider the symbol 71/2. If fractional exponents
are to follow the same rules as integer exponents, the
square of 71/2 must be 7, because
(71/2)2 = (71/2)2
= 71
= 7
Keep the base and multiply the exponents.
7
Simplify a numerical expression containing a rational exponent
with a numerator of 1
Since (71/2)2 and
both equal 7, we define 71/2 to be
. Similarly, we make these definitions:
and so on.
Rational Exponents
If n is a positive integer greater than 1 and
number, then
is a real
8
Example
Simplify: a. 641/2
b. 641/3
c. (–64)1/3
d. 641/6
Solution:
In each case we will change from rational exponent
notation to radical notation and simplify.
a. 641/2 =
=8
b. 641/3 =
=4
9
Example – Solution
cont’d
c. (–64)1/3 =
= –4
d. 641/6 =
=2
10
2. Simplify a numerical expression containing a
rational exponent with a numerator other than 1
11
Simplify a numerical expression containing a rational exponent
with a numerator other than 1
We can extend the definition of x1/n to cover fractional
exponents for which the numerator is not 1. For example,
because 43/2 can be written as (41/2)3, we have
43/2 = (41/2)3 =
= 23 = 8
Because 43/2 can also be written as (43)1/2, we have
43/2 = (43)1/2 = 641/2 =
=8
In general, xm/n can be written as (x1/n)m or as (xm)1/n.
12
Simplify a numerical expression containing a rational exponent
with a numerator other than 1
Since (x1/n)m =
following definition.
and (xm)1/n =
, we make the
Changing from Rational Exponents to Radicals
If m and n are positive integers, the fraction m/n cannot be
simplified, and x  0 if n is even, then
13
Simplify a numerical expression containing a rational exponent
with a numerator other than 1
Comment
Recall that the radicand of an even root cannot be negative
if we are working with real numbers. In the previous rule,
we could state that “x is nonnegative if n is even” to
emphasize this fact. There are no such restrictions on odd
roots.
14
Example
Simplify: a. 82/3 b. (–27)4/3
Solution:
In each case, we will apply the rule
a.
.
or
15
Example – Solution
b.
cont’d
or
16
Simplify a numerical expression containing a rational exponent
with a numerator other than 1
The work in Example 2 suggests that in order to avoid large
numbers, it is usually easier to take the root of the base
first.
17
3. Apply the rules of exponents to an expression
containing rational exponents
18
Apply the rules of exponents to an expression containing
rational exponents
The familiar rules of exponents are valid for rational
exponents. The next example illustrates the use of each
rule.
19
Example
Use the provided rule to write each expression on the left in
a different form.
Problem
a. 42/541/5 = 42/5+ 1/5
Rule
xmxn = xm +n
= 43/5
b. (52/3)1/2 = 5(2/3)(1/2)
(xm)n = xmn
= 51/3
c. (3x)2/3 = 32/3x2/3
(xy)m = xmym
20
Example
cont’d
d.
e.
f.
g.
21