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Simplifying Radicals
Section 5.3
 Radicals
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Definition
Simplifying
Adding/Subtracting
Multiplying
Dividing
Rationalizing the denominator
Radicals - definitions
The definition of
x is the number that when
multiplied by itself 2 times is x.
4  22  2
16 
x  x x  x
2
Simplifying radicals
Most numbers are not perfect squares, but may have a
factor(s) that is (are) a perfect square(s).
The perfect squares are:
1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, ….
Try these - simplify:
If a radical has a perfect square factor, then we
can pull it out from under the sign.
Ex:
25
5 50
2 2
32
48
x5
72
a 4b 7
98
3
27m3n7 
Adding or Subtracting Radicals
To add or subtract square roots you must have like
radicands (the number under the radical).
2 3 2  4 2
Sometimes you must simplify first:
2  18  2  3 2  4 2
Try These
3 5 5 5 
3 75  48 
4 5  5 18 
3 5 5 5 
2 20  3 80 
Multiplying Radicals
You can multiply any square roots together. Multiply any whole
numbers together and then multiply the numbers under the
radical and reduce.
2  3 2  3 4  3 2  6
3 5  5 2  15 10
Try these:
2 7 5 
(3 12 )( 2 3 ) 
 2
2

3 5  2 10 
2 5 
2

Dividing Radicals
To divide square roots, divide any whole numbers and then divide
the radicals one of two ways:
1) divide the numbers under the radical sign and then take the root,
OR
2) take the root and then divide. Be sure to simplify.
20
20

5
5
20 2 5

2
5
5
or
20
 4 2
5
Try These
1
4
40
10
100
4
8 14
2 2
100
25
80
10
Rationalizing Radicals
 It is good practice to eliminate radicals from the denominator of
an expression.
 For example:
3
We need to eliminate
2
2
 We do not want to change the value of the expression, so we
need to multiply the fraction by 1. But “1” can be written in many
ways…
Since
2  2  2 we will multiply by one where 1 
3
2

2
2

3 2
2 2

3 2
2
2
2
Try These
5
3
2 2
5
3 5
10
5 3
12
2 5
3 8
7
20