MATH 95 LECTURE NOTES
Section 7.1: Radical Expressions and Rational Exponents
Spiral of Theodorus
Square Roots
• The number b is a square root of a if b2 = a.
• The principal square root is the positive square root.
• The square root of a negative number is NOT a real number.
Example 1. List the two distinct square roots of the number 9.
Example 2. Evaluate the following expressions. Simplify each as much as possible.
q
√
√
4
(a)
36
(b)
(c)
0.81
25
1
(d)
√
a2 , a > 0
Math 95 Lecture Notes
Section 7.1: Radical Expressions and Rational Exponents
Cube Roots
• The number b is a cube root of a if b3 = a.
• The cube root of a negative number is a real number.
Example 3. Evaluate the following expressions. Simplify each as much as possible.
(a)
√
3
64
(b)
√
3
−8
(c)
√
3
a6
(d)
q
3
1
27
nth Roots
• The number b is an nth root of a if bn = a.
• An even root of a negative number is NOT a real number.
• Ann odd root of a negative number is a real number.
Example 4. Evaluate the following expressions. Simplify each as much as possible.
(a)
√
4
81
Instructor: A.E.Cary
(b)
√
4
−81
(c)
√
5
32
(d)
√
5
−32
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Math 95 Lecture Notes
Section 7.1: Radical Expressions and Rational Exponents
Rational Exponents (Part I)
Radical expressions can be expressed using rational exponents. More specifically,
√
• a = a1/2
•
√
3
a = a1/3
•
√
n
a = a1/n
where n is a positive integer and the radical expression is a real number.
Example 5. Write each radical expression using a rational exponent.
(a)
√
3
4
(b)
√
19
(c)
√
5
−12
(d)
√
4
x
Rational Exponents (Part II)
An expression with rational exponents can be expressed as a radical expression. More specifically,
am/n =
√
n
am
where m and n are positive integers and the radical expression is a real number.
Example 6. Write each expression in radical form.
(a) 62/3
(c) 85/4
(e) 3x4/7
(b) 205/2
(d) y 7/9
(f) (2x)13/2
Instructor: A.E.Cary
Page 3 of 6
Math 95 Lecture Notes
Section 7.1: Radical Expressions and Rational Exponents
Exponent Properties
an
= an−m
am
• an am = an+m
•
• (an )m
• am/n = am·(1/n) = (am )1/n =
n
• (ab) =
• a−n =
•
a −n
b
an bn
•
1
an
=
•
b n
a
=
bn
an
a n
b
=
√
n
am
an
bn
1
= an
a−n
• a0 = 1, a 6= 0
Example 7. Write each expression in radical form. When necessary, use the rules of exponents first. Simplify
each expression as much as possible. Express your simplified answer in exponential form.
(a) x−1/2
(c) 8−2/3
(e) 12z −7/6
(b) −y −1/3
(d) (23)−4/3
(f) (4x)−2/3
Instructor: A.E.Cary
Page 4 of 6
Math 95 Lecture Notes
Section 7.1: Radical Expressions and Rational Exponents
Example 8. Simplify each expression as much as possible.
(a)
x−1/5
10
(b) x1/2 x3/4
(c)
2x4
x9/2
Instructor: A.E.Cary
(d) 4y −1/6 y 2/3
3/2
(e)
x8
4
(f)
x1/2
y 1/4
!−8
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Math 95 Lecture Notes
Section 7.1: Radical Expressions and Rational Exponents
Example 9. Simplify each radical expression after first re-writing the expression in exponential form. List
each simplification in radical form.
(a)
(b)
√
5
x20
√ 5
3
(c) 5
(d)
√
11
x33
√ √
x4x
A general approach to simplifying expressions with radicals and/or rational exponents:
• Answers are considered fully simplified in both radical form and exponential form.
• If you are given an expression in radical form, state your fully simplified answer in radical form.
You will probably use the rules of exponents in your work; make sure to convert back to radical
form in the end.
• If you are given an expression in exponential form, state your fully simplified answer in
exponential form. Never leave negative exponents in your fully simplified answer.
• Some problems do not ask you to fully simplify, but instead ask your to convert an expression
from one form to another. If this is the case, don’t make the problem more work than necessary!
Instructor: A.E.Cary
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