Section 9.4
Radical Expressions
925
Version: Fall 2007
9.4
Radical Expressions
In the previous two sections, we learned how to multiply and divide square roots.
Specifically, we are now armed with the following two properties.
Property 1.
Let
a
and
b
be any two real nonnegative numbers. Then,
√
a
√
b
=
√
ab,
and, provided
b
= 0
,
√
a
√
b
=
a
b
.
In this section, we will simplify a number of more extensive expressions containing
square roots, particularly those that are fundamental to your work in future mathe
matics courses.
Let’s begin by building some fundamental skills.
The Associative Property
We recall the associative property of multiplication.
Associative Property of Multiplication.
Let
a
,
b
, and
c
be any real numbers.
The
associative property of multiplication
states that
(
ab
)
c
=
a
(
bc
)
.
(2)
Note that the order of the numbers on each side of
equation (2)
has not changed.
The numbers on each side of the equation are in the order
a
,
b
, and then
c
.
However, the grouping has changed. On the left, the parentheses around the product
of
a
and
b
instruct us to perform that product first, then multiply the result by
c
. On the
right, the grouping is different; the parentheses around
b
and
c
instruct us to perform
that product first, then multiply by
a
. The key point to understand is the fact that
the different groupings make no difference. We get the same answer in either case.
For example, consider the product
2
·
3
·
4
. If we multiply 2 and 3 first, then multiply
the result by 4, we get
(2
·
3)
·
4 = 6
·
4 = 24
.
On the other hand, if we multiply 3 and 4 first, then multiply the result by 2, we get
2
·
(3
·
4) = 2
·
12 = 24
.
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926
Chapter 9
Radical Functions
Version: Fall 2007
Note that we get the same result in either case. That is,
(2
·
3)
·
4 = 2
·
(3
·
4)
.
The associative property, seemingly trivial, takes on an extra level of sophistication
if we apply it to expressions containing radicals. Let’s look at an example.
l⚏
Example 3.
Simplify the expression
3(2
√
5)
. Place your answer in simple radical
form.
Currently, the parentheses around 2 and
√
5
require that we multiply those two
numbers first. However, the associative property of multiplication allows us to regroup,
placing the parentheses around 3 and 2, multiplying those two numbers first, then
multiplying the result by
√
5
. We arrange the work as follows.
3(2
√
5) = (3
·
2)
√
5 = 6
√
5
.
Readers should note the similarity to a very familiar manipulation.
3(2
x
) = (3
·
2)
x
= 6
x
In practice, when we became confident with this regrouping, we began to skip the
intermediate step and simply state that
3(2
x
) = 6
x
. In a similar vein, once you become
confident with regrouping, you should simply state that
3(2
√
5) = 6
√
5
. If called upon
to explain your answer, you must be ready to explain how you regrouped according to
the associative property of multiplication. Similarly,
−
4(5
√
7) =
−
20
√
7
,
12(5
√
11) = 60
√
11
,
and
−
5(
−
3
√
3) = 15
√
3
.
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 Addition, Elementary arithmetic, Greatest common divisor, radical expressions, simple radical form

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