Simplifying Radicals
Perfect Squares
1
64
225
4
81
256
9
16
100
121
289
25
36
49
144
169
196
400
324
625
4
=2
16
=4
25
=5
100
= 10
144
= 12
LEAVE IN RADICAL FORM
Perfect Square Factor * Other Factor
8
=
4*2 =
2 2
20
=
4*5
=
2 5
32
=
16 * 2 =
4 2
75
=
25 * 3 =
5 3
40
=
4 *10 = 2 10
LEAVE IN RADICAL FORM
Perfect Square Factor * Other Factor
48
=
16 * 3 =
4 3
80
=
16 * 5 =
4 5
50
=
25 * 2 = 25 2
125
=
25 * 5 =
450
=
225 * 2 = 15 2
5 5
+
To combine radicals: combine
the coefficients of like radicals
Simplify each expression
6 7 5 7 3 7 
8 7
5 6 3 7 4 7 2 6 
3 6 7 7
Simplify each expression: Simplify each radical first and
then combine.
2 50  3 32  2 25 * 2  3 16 * 2 
2 *5 2  3* 4 2 
10 2  12 2 
2 2
Simplify each expression: Simplify each radical first and
then combine.
3 27  5 48  3 9 * 3  5 16 * 3 
3* 3 3  5 * 4 3 
9 2  20 2 
29 2
LEAVE IN RADICAL FORM
Perfect Square Factor * Other Factor
18
=
=
288
=
=
75
=
=
24
=
=
72
=
=
Simplify each expression
6 5 5 6 3 6 
3 24  7 54 
2 8  7 32 
Simplify each expression
6 5  5 20 
18  7 32 
2 28  7  6 63 
*
To multiply radicals: multiply the
coefficients and then multiply
the radicands and then simplify
the remaining radicals.
Multiply and then simplify
5 * 35  175  25 * 7  5 7
2 8 * 3 7  6 56  6 4 *14 
6 * 2 14  12 14
2 5 * 4 20  20 100  20 *10  200
 5
2

5* 5 
25 

7* 7 
49  7

8* 8 
64  8

x* x 
x 
 7
2
 8
2
 x
2
2
5
x
To divide radicals:
divide the
coefficients, divide
the radicands if
possible, and
rationalize the
denominator so that
no radical remains in
the denominator
56

7
8
4* 2  2 2
This cannot be
divided which leaves
the radical in the
denominator. We do
not leave radicals in
the denominator. So
we need to
rationalize by
multiplying the
fraction by something
so we can eliminate
the radical in the
denominator.
6

7
6
*
7
42

49
7

7
42
7
42 cannot be
simplified, so we are
finished.
This can be divided
which leaves the
radical in the
denominator. We do
not leave radicals in
the denominator. So
we need to
rationalize by
multiplying the
fraction by something
so we can eliminate
the radical in the
denominator.
5

10
1
*
2
2
10
2

2
This cannot be
divided which leaves
the radical in the
denominator. We do
not leave radicals in
the denominator. So
we need to
rationalize by
multiplying the
fraction by something
so we can eliminate
the radical in the
denominator.
3

12
3
*
12
3

3
3 3

36
Reduce
the
fraction.
3 3

6
3
6
X
Y
4
2
=X
= Y3
6
6
2
P X Y
4
4X Y
8
2
= P2X3Y
= 2X2Y
10 =
25C D
5C4D10
X
3
=
X
=
Y
5
2
X
*X
X
=
Y
=
2
Y
4
Y
Y
3
PX Y
3
=
2
X Y
= XY
7
12 X Y
8
2
=
Y
9
25C D =
Y
5
5
2
* PXY
PXY