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ALGEBRA
CHAPTER 11-1
NOTES
I. Terms to know:
Radical: Also known as the “square root sign”.
 
Perfect square(s): Numbers that have a whole number for a root. Examples are 0, 1, 4, 9,
16, 25, . . .
*** Memorize the squares 1 – 20 and 25
II. To simplify a radical expression:
1. Think of the term under the radical as the product of two numbers, one of which is a
perfect square. (You do not need to find all the factors. Just find which perfect
square(s) would go into the number.)
2. Take the square root of the perfect square and write it in front of the radical symbol.
3. Leave the factor that is not a perfect square under the radical.
4. If the number under the radical is not simplified, do the process again. This will
happen if you do not find the largest perfect square.
Example:
125  25 5  25
Simplify.
108
 36 3
6 3
294
 49 6
7 6
5 5 5
3 128
 3 64 2
On this problem, you need to multiply the 3 and 8.
38 2
 24 2
16 x 4
 16 x 4
 4x
When you have a variable term, take ½ of the exponent (if even).
2
2 18a 6
 2 9 2 a 6
 2 3 a 3 2
 6a 3 2
2b 484b 2
 2b 121 4 b
 2b 11 2 b
 44b 2
2
In this problem, when simplifying, there
were two perfect squares under the
radical (121 & 4). Take the square root
of both.
III. Multiplying radicals.
To multiply radicals, put all expressions under one radical symbol. Multiply, then
simplify if needed.
13
52
13
 13 52
Another way 
 676
 26
 13 13 4
 13 13 4
 13 2  26
4 2a 3 6a 2
 4 2 a 3 2 3 a2
43 2 23a a
 12 2 a
3
 24a 3
75 5 x 3 8 x
 75 3 40 x 2
 225
52
4 10 x 2
 225 2 x 10
 450 x 10
IV. Division with radicals:
Take the root of both the numerator and denominator, then simplify if needed.
7
7
7


36
6
36
Divide.
75
16t 2



49
x4
75
16t 2

25 3
4 4 t2

5 3
4t
49
x4
7
x2
120
10


120
10
 12
 43
2 3
OR 
4 30
10
2 30
10
 30 
 2 

 10 
2 3
Much more complicated!!
Simplify the fraction first, if possible.
75 x5
48 x
25 x 4
16
2
5x
=
4

Simplify the fraction. (  3)
V. Rationalizing the denominator.
3
3

3
3
3 3 3 3 3


 3
3
3
9
5
18t 2

7m
10
5
9 2 t2
5

3t 2

5
3t 2
2
2
10
10


3t 2
6t
Be careful simplifying these!! You
cannot take out a common 2 in the
answer. The numerator is not 10 but
10  3.1622776...

7m
10

7m
10

70m
10
10
10
You cannot divide the 70m by 10 since
the 70m is under the radical