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Manipulating radicals. Manipulating radicals. and. Also:. You should also remember that, by definition, √ a means the positive square root of a. When working with radicals it is important to remember the following two rules:. Simplifying radicals. - PowerPoint PPT Presentation
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© Boardworks Ltd 20101 of 13
Manipulating radicalsManipulating radicals
© Boardworks Ltd 20102 of 13
Manipulating radicals
When working with radicals it is important to remember the following two rules:
You should also remember that, by definition, √a means the positive square root of a.
a
a a
b b=
and
ab a b= ×
Also: × =a a a
© Boardworks Ltd 20103 of 13
Simplifying radicals
Start by finding the largest square number that divides into 50.
This is 25. We can use this to write:
We can do this using the fact that For example:
ab a b= × .
= 5 2
50 = 25×2
= 25 × 2
We are often required to simplify radicals by writing them in the form .a b
Simplify by writing it in the form 50 .a b
© Boardworks Ltd 20104 of 13
= 9 × 5
Simplifying radicals
Simplify the following radicals by writing them in the form a√b.
= 3 5
45 = 9×5
1) 45 2) 98
= 7 2
98 = 49×2
= 49 × 2
.a b
© Boardworks Ltd 20105 of 13
Adding and subtracting radicals
Radicals can be added or subtracted if the number under the square root sign is the same. For example:
Simplify 45 + 80.
Start by writing and in their simplest forms.45 80
= 3 5
45 = 9×5
= 9 × 5
= 4 5
80 = 16×5
= 16 × 5
45 + 80 = 3 5 + 4 5 = 7 5
© Boardworks Ltd 20106 of 13
Multiplying binomials containing radicals
Simplify the following:
1) (4 2)(1+ 3 2) 2) ( 7 2)( 7 + 2)
+12 2 2 6= 4
=11 2 2
+ 2 7 2 7 2= 7
= 5
Problem 2) demonstrates the fact that (a – b)(a + b) = a2 – b2.
In general:
( )( + )a b a b a b
© Boardworks Ltd 20107 of 13
Rationalizing the denominator
When a fraction contains a radical as the denominator we usually rewrite it so that the denominator is a rational number.
This is called rationalizing the denominator. For example:
Simplify the fraction .5
2
In this example we rationalize the denominator by multiplying the numerator and the denominator by 2.
5=
2 25 2
× 2
× 2
© Boardworks Ltd 20108 of 13
Rationalizing the denominator
Simplify the following fractions by rationalizing their denominators.
3)3
4 71)
2
32)
5
2
3
4=
7
× 7
× 7
× 5
× 5
× 3
× 3
2=
3 32 3
=5
25
10
283 7
© Boardworks Ltd 20109 of 13
Rationalizing the denominator
When the denominator involves sums of differences between radicals we can use the fact that
(a – b)(a + b) = a2 – b2
to rationalize the denominator. For example:
Simplify1
.5 2
1 1 5 + 2
= ×5 2 5 2 5 + 2
5 + 2=
5 4
= 5 + 2
© Boardworks Ltd 201010 of 13
Working:
2 3 1 3 1
Rationalizing the denominator
More difficult examples may include radicals in both the numerator and the denominator. For example:
Simplify2 3 1
.3 +1
( 32 3 1 1)
( 3 1
(2 3 1)=
3 +1 ( 3 +1) )
7 3 3=
3 1
= 6 2 3 3 + 1
= 7 3 37 3 3=
2
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