Operations on Radicals

1. Simplifying Radicals

2. Perfect Squares
1
4
9
16
25
36
49
64
81
100
121
144
169
196
225
256
324
400
625
289

3. 4
16
25
100
144
= 2
= 4
= 5
= 10
= 12

4. 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
LEAVEINRADICALFORM

5. 48
80
50
125
450
=
=
=
=
=
3*16
5*16
2*25
5*25
2*225
=
=
=
=
=
34
54
25
55
215
Perfect Square Factor * Other Factor
LEAVEINRADICALFORM

6. +
To combine radicals: combine
the coefficients of like radicals

7. Simplify each expression
=−+ 737576 78
=−++ 62747365 7763 +

8. Simplify each expression: Simplify each radical first and
then combine.
=− 323502
22
212210
24*325*2
2*1632*252
−
=−
=−
=−

9. Simplify each expression: Simplify each radical first and
then combine.
=+ 485273
329
32039
34*533*3
3*1653*93
=+
=+
=+

10. 18
288
75
24
72
=
=
=
=
=
=
=
=
=
=
Perfect Square Factor * Other Factor
LEAVEINRADICALFORM

11. Simplify each expression
=−+ 636556
=+ 547243
=− 32782

12. Simplify each expression
=+ 20556
=+ 32718
=+− 6367282

13. *
To multiply radicals: multiply the
coefficients and then multiply
the radicands and then simplify
the remaining radicals.

14. =35*5 =175 =7*25 75
Multiply and then simplify
=73*82 =566 =14*46
=142*6 1412
=204*52 =10020 20010*20 =

15. ( ) =
2
5 =5*5 =25 5
( ) =
2
7 =7*7 =49 7
( ) =
2
8 =8*8 =64 8
( ) =
2
x =xx * =2
x x

16. To divide radicals:
divide the
coefficients, divide
the radicands if
possible, and
rationalize the
denominator so that
no radical remains in
the denominator

17. =
7
56
=8 =2*4 22

18. =
7
6
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.
=
7
7
*
7
6
=
49
42
7
42
42 cannot be
simplified, so we are
finished.

19. 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

20. 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
3
Reduce
the
fraction.

21. 2
X
6
Y
264
YXP
24
4 YX
108
25 DC
= X
= Y3
= P2
X3
Y
= 2X2
Y
= 5C4
D10

22. 3
X
XX
=
=
XX *2
YY 45
Y
=
= YY 2

23. 33
YPX
27
12 YX
98
25 DC
=
=
= 5
Y
PXYYX *22
5
Y
PXYXY=