Algebra 1 ~ Chapter 11.1
Simplifying Radical
Expressions
Warm Up – Simplify each expression
a.) √9 =
b.) √100 =
c.) √36 =
d.) √49 =
e.) √1 =
List of the first 15 Perfect
Squares
You MUST know this list!!
1
36
121
4
49
144
9
64
169
16
81
196
25
100
225
 An expression that contains a radical
sign (√ ) is a radical expression.
 There are many types of radical
expressions (such as square roots, cube
roots, fourth roots, and so on), but in this
chapter, you will study radical expressions
that contain only square roots.
 The expression under a radical sign is the
radicand.
* A radicand may contain numbers,
variables, or both.
Simplest Radical Form – An
expression containing square root is
in simplest form when…
 the radicand has no perfect square
factors other than 1
 the radicand has no fractions
 there are no square roots in any
denominator.
Ex. 1: Simplify each expression
a.) √24
b.) √45
= √4 · √6
= √9 · √5
= 2√6
= 3√5
Simplest radical
form!
Example 1C – Simplify the radical
expression
32
 16  2
4 2
Factor the
radicand
using
perfect
squares.
32
 4 8
2 8
 2 4  2
 22 2
4 2
Ex. 1 – Simplify the radical expression
d.)
2 125
 2( 25  5 )
e.)
 4 44
 4( 4  11)
 2 5 5
 4 2 11
 10 5
 8 11
Ex. 2 – Simplify each radical
expression
a.) √12
b.) √90
c.) 2√36
d.) √75
e.) √147
f.) -1√52
Ex. 3 – Simplify the radical expression
10  6
 60
 4  15
 2 15
Perform the indicated
operation on the radicals.
Leave answer in simplest
radical form.
Ex. 4 – Simplify each expression
a.) √3 · √15
b.) √2 · √24
c.) 2√3 · 3√3
d.) 4√5 · 2√6
Simplifying Square Roots with
Variables
Remember,
16  4
x  x
2
x  x2
4
x  x3
6
x  x4
8
x  x5
10
x  √x
x  x√x
3
x  x2√x
5
x  x3√x
7
x  x4√x
9
Ex. 5A – Simplify
3
16 x y
4
 ( 16 )( x  x )( y )
2
4
Factor the
radicand
using perfect
squares.
 (4)( x x )( y )
2
 4 xy
2
x
Simplify and rewrite
with #s first, then
radicals.
Ex. 5B – Simplify
3 4
80a b c
 ( 16  5 )( a )( b )( c )
3
4
Factor the
radicand
using perfect
squares.
 (4 5 )( a a )(b )( c )
2
 4ab
2
5ac
Simplify and rewrite
with #s first, then
radicals.
Ex. 6 – Simplify
a.)
2 6
25a b
b.)
28x
3
c.)
2 3 4
300r s t