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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?
31
(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
41
by the same number.
123
3431
and the numerator
By what number
to a rational number? to change it
31
3can we multiply
The answer is . . .
31
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