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(male narrator) In this video,
we will take a look at rationalizing denominators
that have complex binomial expressions in them.
You may recall, similar to other radicals,
we can rationalize a binomial denominator
by multiplying by the conjugate.
The conjugate is made up of the same terms,
where one has a positive in the middle
and one has a negative in the middle.
You also recall from multiplying sums and difference
that we only need to multiply
the first two terms and the last two terms--
as the middle terms will subtract out to 0:
a times a would be a squared,
and bi times bi would be -b squared, i squared.
However, remember i squared is -1,
which will change the negative into a positive.
Let's take a look at some examples where we rationalize
by multiplying by this conjugate.
In this problem, we've got a binomial with two terms,
so we'll multiply by the conjugate:
2 plus 5i, in both the numerator and denominator.
In the numerator, we simply have to distribute:
4i times 2 is 8i,
and 4i times 5i is +20, i squared.
However, recall that i squared is -1,
which changes the +20 into a -20.
In the denominator,
we only have to multiply the first and last terms:
2 times 2 is 4,
and -5i times +5i is -25, i squared.
However, the i squared changes the negative into a positive,
because i squared is -1.
We now have 4 plus 25, which is 29.
We could factor the numerator to see if it reduces
by pulling out a factor of 4, leaving 2i minus 5 over 29.
But we'll find that this does not reduce.
Either of these two answers
would be considered the correct, simplified solution.
Let's take a look at one more example
where we're asked to rationalize the denominator
by multiplying by the conjugate.
In this problem, we have 3 plus 5i in the denominator.
The conjugate would be 3 minus 5i,
and this is what we must multiply by
in both the numerator and the denominator.
Remember, we must multiply the entire numerator
by the 3 minus 5i,
which will require us to use FOIL:
4 times 3 is 12, and 4 times -5i is -20i;
-2i times 3 is -6i;
and -2i times -5i is +10, i squared.
However, remember i squared is -1,
which will change the +10 to a -10.
Let's go ahead and combine these like terms:
12 minus 10 is going to be 2;
-20i minus 6i is -26i;
over...in our denominator, we need to multiply
the first and last terms together:
3 times 3 is 9,
and 5i times -5i is -25, i squared.
However, because i squared is -1,
that changes the -25 to a +25: 9 plus 25 is 34.
This is our denominator.
To see if we can reduce,
we'll factor a 2 out of the numerator,
making it 1 minus 13i over 34.
Sure enough, 2 and 34 have a common factor of 2.
This gives us our final answer: 1 minus 13i over 17.
Just as with regular radicals,
if there is a binomial in the denominator,
we clear the radical-- or the i--
by multiplying by the conjugate.
