Download Section 8.2 Multiplying, Dividing, and Simplifying Radicals

Page 1 of 6
Section 8.2
Multiplying, Dividing, and Simplifying Radicals
Observe the following:
√
√
and
Product Rule
For nonnegative real numbers
√
√
and ,
√
and
√
WARNING: The rule does not apply to sums. In general, √
Example 1
Find each product. Assume that
(a) √
√
√
√
.
(b) √
√
(c) √
√
A square root radical is simplified when no perfect square factor remains under the radical sign.
We accomplish this by using the product rule in the form: √
√ √
Example 2
Simplify each radical.
(a) √
(Method #1: Identify the greatest perfect square factor)
(b)
√
(c) √
Page 2 of 6
Example 3
Simplify each radical.
(Method #2: Use a factor tree to write the prime factorization.)
(a) √
(b)
√
(c)
√
Example 4
Find each product and simplify.
(a) √
√
(b) √
√
Page 3 of 6
Quotient Rule
For nonnegative real numbers
√
and ,
and
,
√
√
Example 5
Simplify each radical.
(a)
48
3
(c)
5
36
(b)
4
49
Example 6
8 50
Simplify
4 5
Some problems require both the product and quotient rules.
Example 7
Simplify
3 7

8 2
Page 4 of 6
Radicals can involve variables. Simplifying such radicals can get a little tricky.
If
represents a nonnegative number , then √
If
represents a negative number , then √
For any real number ,
√
To avoid negative radicands, variables under radical signs will be assumed to be nonnegative in
this course. Therefore, absolute value bars are not necessary (in this course).
Example 8
Simplify each radical. Assume that all variables represent positive real numbers.
(a) √
(b) √
√
(d) √
(c)
(e) √
(f) √
Page 5 of 6
In general,
n
a n b 
Example 9
Simplify each radical.
(a)
3
108
(b)
4
160
(c)
4
16
625
n
and
n
a

b
Page 6 of 6
To simplify cube roots with variables, use the fact that for any real number a,
a3  a
This is true whether a is positive or negative.
3
Example 10
Simplify each radical.
(a)
3
(c)
3
z9
54t
5
(b)
3
8x 6
(d)
3
a15
64