Download Rational Exponents

Name __________________________________
Period __________
Date:
Essential Question: How is a rational exponent related to a
radical?
10-1
Rational
Exponents
Topic:
Standard: N-RN.1
Objective:
Explain how the definition of the meaning of rational exponents follows
from extending the properties of integer exponents to those values,
allowing for a notation for radicals in terms of rational exponents.
To extend the meaning of exponents to include rational
numbers.
You have studied integral exponents, and you know the
meaning of the expression
when a is an integer. For
example,
Rational Number
Exponents:
But, what if a is not an integer? What if a is a rational
number?
⁄
Let
(
Then
.
⁄
)
By the definition of square root,
√

Summary
⁄
√
(
)
Similarly,
⁄
√
⁄
Now consider
(
Then
⁄
(
)
)
By the definition of cube root,
√
(
Moreover,
⁄

√
⁄
)
(√ )
(√ )
Definition of bp/q
If p and q are integers with
number, then
⁄
Example 1:
, and b is a positive real
(√ )
√
⁄
Simplify
Solution 1
⁄
(√
Solution 2
⁄
√
)
√
Notice that the first computation is easier than the second.
This is usually the case.
2
Exercise 1:
⁄
Simplify
Calculators are especially useful for evaluating rational
exponent expressions. The figure below is a screen capture of
a graphing calculator evaluating the above expression.
Example 2:
Simplify
a.
⁄
⁄
(√
)
3
b.
⁄
(√ )
Exercise 2:
Simplify
a.
⁄
b.
4
Laws of Exponents:
The Laws of Exponents also apply to powers with rational
exponents.
Let m and n be rational numbers and a and b be positive real
numbers, with a  0 and b  0 when they are divisors. Then,
1. aman = am+n
2. (ab)m = am bm
3. (am)n = amn
4.
5.
6. ( )
These laws can be used with the definition of rational
exponents to simplify many radical expressions. The first step
is to write the expression in exponential form.
Exponential Form:
Example 3:
An expression is in exponential form if it is a power or a
product of powers and contains no radicals.
a. Write √
in exponential form.
⁄
√
(
)
⁄
⁄
⁄
⁄
⁄
5
b. Simplify (
√
⁄
)
.
⁄
(
√
)
⁄
(
(
)
⁄ )(
⁄
⁄ )
⁄
√
Exercise 3:
a. Write
√
b. Simplify (
in exponential form.
)
⁄
6
Simplest Radical
Form:
An expression containing nth roots is in simplest radical form if:
1. No radicand contains a factor (other than 1) that is a perfect nth
power.
2. Every denominator has been rationalized. Specifically, no
radicand is a fraction, and no denominator contains a radical.
Write √
Example 4:
√
√ in exponential form and in simplest radical form.
√
√
√
⁄
⁄
⁄
⁄
Exponential
Form
⁄
⁄
⁄
⁄
⁄
Simplest
Radical
Form
Exercise 4:
√
Express √
in simplest radical form. (Hint: First find the
√
prime factorizations of 128 and 256.)
7
Example 5:
Solve:
(
a.
)
)
[(
⁄
]
⁄
⁄
⁄
⁄
b.
⁄
(
Exercise 5:
⁄
)
Solve:
a.
⁄
8
b. ( )
⁄
c. (
)
d. (
)
⁄
⁄
Class work:
p 457 Oral Exercises: 1-30
Homework:
p 458 Written Exercises: 1-35 odd
p 458 Mixed Review: 1-6
p 458 Written Exercises: 37-50 odd
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