Avon High School
Section: P.3
ACE COLLEGE ALGEBRA II - NOTES
Radicals and Rational Exponents
Mr. Record: Room ALC-129
Day 3
Square Roots
Definition of the Principal Square Root
If a is a nonnegative real number, the
nonnegative number b such that b 2  a ,
denoted by b  a is the principal square
root of a.
Example 1
a.
c.
Evaluate.
 9
9  16
b.
1
25
d.
9  16
Investigation:
How would you evaluate the following:
-25 ?
Try entering it into your calculator.
The Square Root of a
2
Simplifying a 2
For any real number a,
a2  a
In other words, the principal square root
of a 2 is the absolute value of a.
Simplifying Products and Quotients of Square Roots
Example 2
Simplify.
a.
48
b.
5 x  10 x
c.
25
16
d.
150 x 3
2x
Simplifying Sums and Differences of Square Roots
Example 3
a.
Simplify.
5 27  12
b.
7 x  98 x  2 x  5 28 x
Rationalizing Denominators
This is a process you learned in Algebra I which involves rewriting a radical expression as an equivalent
expression in which the denominator no longer contains a radical.
Example 4
a.
Simplify.
15
6
b.
6
12
Conjugates
Radical expressions that involve the sum and difference of the same two
terms are called conjugates.
a  b and a  b are conjugates.
Multiplying Radical Conjugates

Example 5
a.
7
5 3
a b

  a  b
a b 
2
2
 a b
Simplify.
b.
8
3 2 4
Note:
n
is called the radical.
The expression under the radical is
the radicand.
Other Kinds of Roots
The Principal nth Root of a Real Number
n
a  b means that b n  a
If n, the index, is even, then a is nonnegative (a  0) and b is also
nonnegative (b  0) . If n is odd, a and b can be any real number.
Fourth Roots
1
4
16
1
2
4
81
3
4
256
4
4
625
5
4
Simplifying, Multiplying and Dividing Higher Roots
Example 6
a.
3
40
Simplify.
b.
5
85 8
c. 3 3 81  4 3 3
Cube Roots
1
3
8
1
2
Fifth Roots
3
27
3
1
2
3
64
4
1
5
32
5
3
3
125
5
3
216
6
3
343
7
3
Rational Exponents
a1/n
The Definition of
If n a represents a real number, where n  2 is an integer, then
1
m
a n  n a and furthermore, a n 
Example 7
 a
n
m

 a  for any m.
m
Simplify.
1
4
a. 256 4
b. 27 3
 4  8 
d.  2 x 3  5 x 3 



n
e.
20 x 4
5x
3
2
c. (32)
f.
9
x3

2
5
5
243