Simplifying Radicals
Perfect Squares
1
4
9
16
25
36
49
64
81
100
121
144
169
196
225
256
324
400
625
289
4
16
25
100
144
= 2
= 4
= 5
= 10
= 12
8
20
32
75
40
=
=
=
=
=
2*4
5*4
2*16
3*25
10*4
=
=
=
=
=
22
52
24
35
102
Perfect Square Factor * Other Factor
LE
AV
E I
N R
AD
ICA
L F
OR
M
48
80
50
125
450
=
=
=
=
=
3*16
5*16
2*25
5*25
2*225
=
=
=
=
=
34
54
225
55
215
Perfect Square Factor * Other Factor
LE
AV
E I
N R
AD
ICA
L F
OR
M
+To combine radicals: combine
the coefficients of like radicals
Simplify each expression
737576 78
62747365 7763
Simplify each expression: Simplify each radical first and then combine.
323502
22
212210
24*325*2
2*1632*252
Simplify each expression: Simplify each radical first and then combine.
485273
229
22029
34*533*3
3*1653*93
18
288
75
24
72
=
=
=
=
=
=
=
=
=
=
Perfect Square Factor * Other Factor
LE
AV
E I
N R
AD
ICA
L F
OR
M
Simplify each expression
636556
547243
32782
*To multiply radicals: multiply the
coefficients and then multiply
the radicands and then simplify
the remaining radicals.
35*5 175 7*25 75
Multiply and then simplify
73*82 566 14*46
142*6 1412
204*52 10020 20010*20
2
5 5*5 25 5
2
7 7*7 49 7
2
8 8*8 64 8
2
x xx * 2x x
To divide radicals:
divide the
coefficients, divide
the radicands if
possible, and
rationalize the
denominator so that
no radical remains in
the denominator
7
568 2*4 22
7
6This 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.
7
7*
7
6
49
42
7
42
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.
10
5
2
2*
2
1
10
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.
12
3
3
3*
12
3
36
33
6
33
6
3Reduce
the
fraction.
2X
6Y
264 YXP
244 YX
10825 DC
= X
= Y3
= P2X3Y
= 2X2Y
= 5C4D10
3X
XX
=
=
XX *2
YY 45Y
=
= YY 2
33YPX
2712 YX
9825 DC
=
=
= 5Y
PXYYX *22
5Y
PXYXY=