Assalamualaikum warahmatullahi wabaraktuh
In this video, we will see how to
rationalize surds.
Rationalize means to make a fraction has no surd term in its denominator.
The first one is to rationalize by
conjugate.
The second one is to rationalize by
itself.
Let's look at the first case first.
We are going to rationalize by conjugate
if the fraction has more than one term.
In each denominator with at least one of
the term is a surd.
For example, 1/(a +/- sqrt(b), or we can have it as 1/[sqrt(a) +/- sqrt(b)]
Conjugate in this context means pair.
So to rationalize this type of surd.
We need to find it spare first. Generally
to find the conjugate of surds
we just need to change the surd sign.
Take a look at this form first focus on
its denominator in the denominator.
We have ‘a +/- sqrt(b)’. Since to find conjugate.
We just need to change this surd sign.
Therefore the conjugate for this one is ‘a -/+ sqrt(b)’,
but what if both of the terms are surds. Do we need to change of their signs?
The answer is NO. We just need to change either one of the signs.
For example here
the denominator is sqrt(a) +/- sqrt(b).
Therefore the conjugate for this one is
sqrt(a) -/+ sqrt(b)
the sign for sqrt(a) didn't change.
We just change the sign for sqrt(b). We’ll see more of this in example C & D.
The second type is to rationalize by
itself.
We are going to use the second type if
you only have surd in the denominator.
For example, we can have it in the form
of 1/sqrt(a).
So to rationalize this we need to multiply
this by itself that is sqrt(a).
Now, let's rationalize.
We have 4-sqrt(2) in its denominator. To rationalize this we need
to find its conjugate first.
And as I said earlier to find its
conjugate, you just need to change the
sign for its surds.
So the conjugate for this one is 4+sqrt(2).
Now to rationalize you need to multiply
the original fraction with conjugate over
conjugate that see this example again.
So this is the original fraction.
We have 3/(4-sqrt(2)),
so to rationalize this we need to
multiply by its conjugate over conjugate
and we already know that the conjugate
for this one is 4+sqrt(2) / 4+sqrt(2)
The next thing that we're going to do is
to multiply and simplify them.
So let's multiply first.
Multiplying this you are going to get
12+3sqrt(2) / [16+4sqrt(2)-4sqrt(2)-2]
Simplify more.
So 12+3sqrt(2) / 14,
and since we cannot simplify any more
than this, this should be our final
answer. Moving on to the next one here.
We have surd in its denominator.
Therefore, we need to rationalize
this again.
And as I said before to rationalize you
need to multiply this by its conjugate
over conjugate, let’s write the
conjugate first. To find the conjugate.
We need to change the sign of surds.
Therefore, the conjugate for this one will be
-sqrt(3)-2
Now, let's rationalize this
[2/(sqrt(3) -2)] x [(-sqrt(3) -2) / (-sqrt(3) - 2)]
and then we are going to expand and simplify.
I'm going to fast forward this until the
last answer. You can pause this video if
it's too fast for you and
our final answer should be -2sqrt(3)-4
Very well.
Let's proceed in question c
you can see we have two surds in its
denominator. To find its conjugate.
We just need to change either one of the
signs.
So we have two possible conjugates here.
We could either have -sqrt(3)+sqrt(2) or sqrt(3)-sqrt(2)
Both conjugates are correct.
I'm going to choose this one as my
conjugate and again to rationalize this
you need to multiply the original
fraction,
[1/(sqrt(3)+sqrt(2))] x [(sqrt(3)-sqrt(2)) / (sqrt(3)-sqrt(2))]
The next thing that we're going to to do is to expand and simplify.
sqrt(3)-sqrt(2) over to expand this one.
I'm going to use a trick here.
You can also your lecturer in your
tutorial session about this trick.
So expanding these two here will become 3-2.
and our final answer should be sqrt(3)-sqrt(2).
Now, let me explain question e first and then you can do question d.
and f on your own now.
Let's see e first. Question e we only
have surd in its denominator.
So to rationalize this we need to
multiply this by itself.
This belongs to case number two. To
rationalize case number two
we are going to multiply this by itself.
So we have 2/sqrt(2), we
are going to multiply this by itself.
So (sqrt(2)) / (sqrt(2)) and as before expand and simplify,
so we are going to get (2sqrt(2))/2
and you know that we can cancel this 2.
So our final answer should be sqrt(2).
Okay then, pause this video now,
we will discuss the answer for question d
and question f. Finished.
Let's check your answers.
So in question d, you can have two
conjugates here either
-sqrt(5)-sqrt(3) or sqrt(5)+sqrt(3)
For this solution, I'm going to use this as my conjugate.
I'm going to fast forward the solution here.
If you found this too fast for you, you could always pause the video.
After expanding this, you are going get
(-4sqrt(5) -4sqrt(3))/2
This should not be your final answer
because you always need to give your
answer in the most simplified form.
So your final answer should be -2sqrt(5) -2sqrt(3).
Did you get the same answer. Very good
Moving on to question f.
This is very easy to solve.
You just need to multiply this by itself.
So your final answer should be -sqrt(5)/5.
Now that you know all this. You are ready
to answer this question.
Ignore the numerator.
Your goal is to rationalize this fraction.
So focus on its denominator, so let's
find its conjugate first.
You can either have -sqrt(5)+sqrt(2) or sqrt(5)-sqrt(2)
Both will give you the same answers.
Can you do this on your own.
Let's try okay, pause this video now, I
will discuss this after you are finished.
All done.
So first things first multiply the
original with its conjugate over
conjugate.
I choose this one as my conjugate.
If you use this conjugate here, you are
still going to get the same answer.
The next thing that you are going to do
is expand and simplify.
So after expand and simplify you are
going to get (7-2sqrt(10))/3.
Since we cannot simplify more
this one should be your final answer.
Right
So that's it for now. In the next video.
We are going to discuss how to solve
equations involving surds.
So goodbye for now.
Thank you for your attention.
Have a great day and take care.
