9.3
Simplifying Radicals
What is the maximum speed of a sailboat?
Goal
Simplify radical
expressions.
The design of a sailboat
affects its maximum speed.
In Example 4 you will use a
boat’s water line length to
estimate its maximum speed.
Key Words
• radical
• simplest form of a
radical expression
• product property of
radicals
• quotient property of
radicals
The simplest form of a radical expression is an expression that has no perfect
square factors other than 1 in the radicand, no fractions in the radicand, and no
radicals in the denominator of a fraction. Properties of radicals can be used to
simplify expressions that contain radicals.
PRODUCT PROPERTY OF RADICALS
a
b
a
p b
where a ≥ 0 and b ≥ 0 Example: 4
p
5 4
p 5
25
Student Help
STUDY TIP
There can be more
than one way to factor
the radicand. An
efficient method is to
find the largest perfect
square factor. For
example, you can
using
simplify 48
Simplify with the Product Property
Simplify the expression.
a. 5
0
b. 4
8
Solution Look for perfect square factors to remove from the radicand.
a. 5
0 25p2
48
16
p3 16
p 3 43.
1
EXAMPLE
Factor using perfect square factor.
25 p 2
Use product property.
52
Simplify: 2
5
5.
b. 4
8 4p2
1
Factor using perfect square factor.
4p4p3
Factor using perfect square factor.
4 p 3
Use product property.
43
Simplify: 4
2 4.
2
Simplify with the Product Property
Simplify the expression.
1. 1
2
2. 3
2
3. 7
5
9.3
4. 1
80
Simplifying Radicals
511