Rationalizing the denominator

63

26

There is an agreement

31

in mathematics that we don’t leave a radical

in the denominator of a fraction.

So how do we change the denominator of a fraction?

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(Without changing the value of the fraction, of course.)

The same way we change the denominator of any fraction!

(Without changing the value of the fraction, of course.)

41

123

3431

We multiply the denominator

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by the same number.

123

3431

and the numerator

By what number

to a rational number? to change it

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3can we multiply

The answer is . . .

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3

. . . by itself!

3 23 3

Remember,

is the number we squaren

to get n.

we’d better get n.So when we square it,

2n n

In our fraction, to get the radical out of the denominator,

we can multiply numerator and denominator by

.3

31

3

1

3331

33

In our fraction, to get the radical out of the denominator,

we can multiply numerator and denominator by

.3

3

133

Because we are changing the denominator

we call this process rationalizing.

to a rational number,

2

4 22

24

Rationalize the denominator:

2

24

(Don’t forget to sim

plify)

222

12

64

128

1212128

Rationalize the denominator:

36

3

1296

(Don’t forget to sim

plify)

(Don’t forget to sim

plify)

When there is a binomial with a radical in the denominator of a fraction, you find the conjugate and multiply. This gives a rational denominator.Ex: 5 6 Conjugate: 5 6

3 2 2 Conjugate: 3 2 2

What is conjugate of 2 7 3?

Answer: 2 7 3

Simplify: 5 6

5 3

5 6

5 3

5 6

5 3

5 3

5 3Multiply by the conjugate.

5 3 5 6 5 18

5 9

FOIL numerator and denominator.

Next

Simplify

2

5 + 2

2

5 2

5 – 2

5 – 2

·

2(5) – 2 22 25 – ( 2)

10 2 2

25 – 2

10 – 2 2

23=

23 9 5

4Combine like terms

Try this on your own:

6

3 2Answer:

3 6 2 3

7