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Simplifying Radicals
Radicals
2
5
32 6
10
Simplifying Radicals
45 
9 5

9 
 3 5
Express 45 as a product
using a square number
5
Separate the product
Take the square root of
the perfect square
Some Common Examples
12  4 3
 4 3
 2 3
75  25 3
 25  3
5 3
18  9 2
 9 
3 2
2
Harder Example
245
 49 5
 49 
7 5
5
Find a perfect square
number that divides evenly
into 245 by testing 4, 9,
16, 25, 49 (this works)
Addition and Subtraction
You can only add or subtract “like” radicals
5 3 5  4 5
3 7 7  2 7
5 2 3 6 3 2  2 2 3 6
You cannot add or
subtract with
More Adding and Subtracting

75  7 3  8
You must simplify all
radicals before you
can add or subtract
25  3  7 3 
4 2
 5 3  7 3  2 2
 12 3  2 2
Multiplication
Consider each radical as having two
parts. The whole number out the front
and the number under the radical sign.
7 2  3 5  21 10
You multiply the outside numbers
together and you multiply the numbers
under the radical signs together
More Examples
6  5 7  5 42
8 3  2 6  16 18
 16 9  2
 16 3 2
 48 2
Note that 18 can
be simplified
Try These
3 6  4 2  12 12
 12 4  3
 24 3
7 10  3 15
 21 150
 21 25  6
 105 6
Division
As with multiplication, we consider the
two parts of the surd separately.
12 10  3 5
12 10

3 5
10
 4
5
 4 2
Division
8 75  5 3
8 75

5 3
8 75

5 3
8

25
5
8
 5
5
 8
Important Points to Note
ab 
a 
b
a

b
a 
b
However
Radicals can be
separated when you have
multiplication and division
ab 
a 
b
a b 
a 
b
Radicals cannot
be separated
when you have
addition and
subtraction
Rational Denominators
Radicals are irrational. A fraction with a radical in
the denominator should to be changed so that the
denominator is rational.
3
5

3

5
3 5

5
5
5
Here we are
multiplying by 1
The denominator is
now rational
More Rationalising Denominators
6
5 3

6
5 3
6 3

15
2 3

5

3
3
Multiply by 1 in
3
the form 3
Simplify
Review Difference of Squares
2
2
(a  b)( a  b)  a  ab  ab  b
 a 2  b2
When a radical is squared, it is no longer
a radical. It becomes rational. We use
this and the process above to rationalise
the denominators in the following
examples.
More Examples
6
5 3
6
5 3


5 3
5 3
6(5  3 )

25  9
6(5  3 )

16
3(5  3 )

8
Here we multiply by
5 – 3 which is
called the conjugate
of 5 + 3
Simplify
Another Example
1 2

3 7
1 2

3 7
3 7
3 7

3  6  7  14
37

3  6  7  14
4
 
3  6  7  14
4
Here we multiply
by the conjugate
of 3  7 which
is 3  7
Simplify
Try this one
6 5
6 5
2 5  3 The conjugate of


2 5 3
2 5 3
2 5  3 2 5  3 is 2 5  3
2 30  10 5  18  5 3

4 25  9
Simplify
2 30  10 5  3 2  5 3

20  3
2 30  10 5  3 2  5 3

17
See next
slide
Continuing
2 30  10 5  3 2  5 3

17
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