This document discusses operations involving radicals such as square roots. Some key points covered include: 1) Simplifying radicals by pulling out perfect squares and simplifying inside and outside the radical. 2) Multiplying radicals by multiplying only the numbers outside the radicals while keeping numbers inside the same. 3) Rationalizing denominators by multiplying the denominator and numerator by the radical in the denominator to remove radicals from the denominator. 4) Adding and subtracting radicals by combining like terms where only the numbers outside the radicals are added/subtracted and numbers inside are kept the same.

1. Simplifying Radicals
Operations with Radicals

2. 2
2
2
2
2
2
1 1
2 4
3 9
4 16
5 25
6 36
=
=
=
=
=
=
1 1
4 2
9 3
16 4
25 5
36 6
=
=
=
=
=
=
2
2
2
2
2
2
7 49
8 64
9 81
10 100
11 121
12 144
=
=
=
=
=
=
49 7
64 8
81 9
100 10
121 11
144 12
=
=
=
=
=
=

3. 2
1) a a=
b) a2 ba= ×
3)
a
b b
a
=

4. Simplify:
Step 1
Look for Perfect Squares
(Try to use the largest
perfect square possible.)
Step 2
Simplify Perfect Squares
Step 3
Multiply the numbers
inside and outside the
radical separately.
48
3 16
43
•
•
4 3

5. Simplify:
48
4 12
34
2 2 3
•
•
• •
4 3

6. 2
a a=
2
x x=
Any even power is a perfect square.
4 2
10 5
90 45
=
=
=
x x
x x
x x
The square root
exponent is half of
the original exponent.

7. Odd powers
When you take the square root of an odd
power, the result is always an even power and
one variable left inside the radical.
5 2
11 5
91 45
=
=
=
x x x
x x x
x x x

8. Simplifying using variables
When you simplify an even power of a variable
and the result is an odd power, use absolute
value bars to make sure your answer is positive.
14 7
14 12 7 6
=
=
x x
x y x y
Even powers
do not need
absolute
value.

9. Simplify: 3
16x
Step 1
Pull out perfect
squares
Step 2
Simplify
16 2
x x
x4 x• •
4x x

10. a ab b× =
You can only multiply radicals by other radicals
8 3•
Both under the radical
CAN multiply
8 3•
Not under the radical
CANNOT multiply

11. 8 12•Simplify:
Step 1
Are both radicals in
simplest radical form? If
not, Simplify
Step 2
Multiply numbers IN
FRONT of the radical
Multiply numbers
UNDER the radical
Step 3
Put in Simplest Radical
Form.
2 2 2 3•
2 2 32•
4 6

12. Simplify:
( )
2
2 3
Step 1
Rewrite
Step 2
Multiply IN FRONT of
the radical
Multiply UNDER the
radical
Step 3
Simplify
( )( )2 3 2 3
( )( )2 3 2 3
4 9
4 3•
12

13. a
b b
a
=
No radicals (irrational numbers) may remain in the denominator. To
remove them, use a process called rationalizing the denominator

14. 12
49
Simplify:
12
49
Step 1
Do I have a perfect square in the
denominator?
Step 2
Simplify both your numerator
and your denominator
12 2 3=
49 7=
2 3
7

15. Simplify:
Step 1
Do I have a perfect square in the
denominator?
Step 2
Can I reduce my fraction to get a
perfect square in the
denominator?
Step 2
Simplify both your numerator
and your denominator
75
12
75
12
75
12 3÷
3÷
25
45
2

16. Simplify:
Step 1
Identify the radical in the
denominator
Step 2
Multiply denominator AND
numerator by the radical in the
denominator
Step 3
Simplify
3
7 3
7
3
7 7•
7• 21
49
=
21
7

17. Review
2 4x x+ =
Like Terms
6x
We only add the
numbers in front
2
2 2x x+
Not like terms.
The Same is True with Radicals
2 3 4 3+ = 6 3 2 3 2 2+
Like Terms We only add the
numbers in front
CANNOT add
Not like terms.
CANNOT add

18. Step 1
Are both terms in simplest form? If
not, simplify.
Step 2
Are our radicals like terms?
Step 3
Add or subtract the numbers in
FRONT of the radical
Keep the number on the UNDER the
radical the
SAME!!!SAME!!!
4 3 12−
4 3 2 3−
4 3 32−
2 3
Simplify:

19. ( )2 4 12+Simplify
Step 1
Are all terms in simplest form?
Step 2
Distribute
Step 3
Multiply the #’s in front of the
radical
Multiply the #’s under the
radical
Step 4
Simplify and/or Combine like
terms… IF WE CAN
( )2 4 2 3+
( )( )2 4
4 2
( )( )2 2 3+
( )( )2 4 ( )( )2 2 3+
62+

20. You can multiply two radical expressions using
FOIL, the same way you do with binomials.
Example:
( )( )2 5 3 3 5
(2)(3) (2)( 3 5) ( 5)(3) ( 5)( 3 5)
6 6 5 3 5 3 25
6 3 5 3(5)
9 3 5
+ −
+ − + + −
− + −
− −
− −

21. Conjugates are the sum and difference of the same
two terms.
Example:
and
When you multiply conjugates, no radicals are left.
(The middle term cancels out, like in difference of
two squares).
( )5 3+ ( )5 3−