Simplifying Radicals Radicals Simplifying Radicals
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Transcript Simplifying Radicals Radicals Simplifying Radicals
Simplifying Radicals
Radicals
2
5
32 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 2
3 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
37
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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