5.6 Radical Expressions
Objectives:
1. Simplify radical expressions.
2. Add, subtract, multiply and divide
radical expressions.
Simplifying Radials
• Radicals are considered simplified when the
following occurs:
1. The index is as small as possible.
2. The radicand contains no factors other
than 1 that are the nth powers of an integer or
polynomial.
3. The radicand contains no fractions
4. No radicals are in the denominator.
Adding and Subtracting Radicals
• Like algebraic expressions, radical expressions
can only be added or subtracted if the
radicands and indices are the same.
• Sometimes simplifying the radical the radical
will create like-radicals.
5 6 5  3  7 5  3
3 20  4 45  3 4  5  4 9  5
 3  2 5  4  3 5  6 5  12 5
6 5
3
3
3
3
3
Product Property of Radicals
• For any real numbers a and b and any integer
n>1
1. If n is even and a and b are both nonnegative,
n
then
ab  n a  n b
2. If n is odd, then n
n
n
ab  a  b
Example:
32a b  16  2a ab
2 4
 4a b 2a
5 8
4
8
Multiplying Radicals
Radicals with different radicands can be
multiplied. Multiply coefficients and multiply
radicands too.
Example:
(2  3)( 5  2 6)
 2 5  15  4 6  2 18
 2 5  15  4 6  6 3
Quotient Property of Radicals
• For any real numbers a and b≠0 and any
n
integer n>1,
if all roots are
a
a
n
 n
defined.
b
b
3
Example:
3
2
8  8

3
27
3
27
Conjugates
• Conjugates are binomials that are the same
except for the sign between them. Their
product will always be a rational number.
Example: For
the conjugate
3 5
is
3 5
Multiplying them:
( 3  5)( 3  5)
 9  5 3  5 3  25  3  25  22
Rationalizing the Denominator
One radical in the
denominator:
1. Separate radical into
two different radicals
2. Simplify radical if
possible.
3. Multiply numerator and
denominator by radical
in denominator or by
the radical necessary to
get rid of radical.
4. Simplify if necessary.
Radical separated by
addition/subtraction:
1. Multiply numerator and
denominator by
conjugate of
denominator.
2. Simplify, if necessary.
Examples
• Simplify.
5
3 5 3


3
3 3
3
4 3 4 3 25
 3 3
5
25
5
3
3
100
100
 3

5
125
• Simplify
5 6
5 1

5 1
5 1
5 5  5  30  6

25  5  5  1
5 5  5  30  6

4
More Examples
• Simplify
3 45  5 80  4 20
Simplify
3
6 7
4 40a b
 3 9  5  5 16  5  4 4  5  4 8  5a b b
3
 9 5  20 5  8 5
 3 5
6 6
 8a b
2 23
5b
Homework
page 254-255
16-52 even