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Manipulating radicalsManipulating radicals

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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

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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

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= 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

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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

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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

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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

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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

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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

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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