We're asked to rationalize and simplify this expression right over here
and like many problems there are multiple ways to do this.
We could simplify a little bit then rationalize, then simplify a little bit more, or we could just rationalize and simplify.
And just to make sure that you know what they are even talking about rationalize is just a fancy word,
fancy way of saying we don't want to see any square roots of numbers in the denominator. Thats all it says.
So try to get these things outside of the denominator.
So the first thing we can do is just simplify then rationalize, then we can think about other ways to do it.
So what I'd like to do first is say the principle square root of 8 that can be simplified a little bit because
8 is the same thing as the square root of 4 times 2 which is the same thing as
the square root of 4 times the square root of 2. So we can rewrite this entire expression as,
the numerator is still the same, 16 plus 2X squared. All of that over, we can rewrite this as the square root of 4 times the square root of 2.
The principle square root of 4 we know is just 2. So the square root of 8 we can rewrite as 2 times the principle square root of two.
And I've simplified a little bit, I've done no rationalizing just yet, and it looks like there is a little more simplification I can do first.
Because everything in the numerator and everything in the denominator is divisible by 2.
So lets divide the numerator by 2.
So if you divide the numerator by 2, 16 divided by 2,
or you could view it as multipying the numerator and denominator by one half.
So 16 times one half is 8.
2X squared times one half is just X squared. And then 2 times the principle square root of 2 times one half is just the square root of 2.
It is 1 square roots of 2.
So this whole thing has simplified to 8 plus X squared, all of that over the square root of 2.
And now lets rationalize this.
The best way to get this radical out of the denominator is just multiply the numerator and the denominator by the principle square root of 2.
So lets do that. So times the principle square root of 2 over the principle square root of 2.
Now just to show that it works on the denominator what is the principle square root of 2 times the principle square root of 2?
Well its going to be 2. And in our numerator, we are going to distribute this term onto both terms in this expression,
so you have 8 times the principle square root of 2 plus the square root of 2 times X squared.
And we could consider this done, we have simplified the expression, or if you want you could break it up.
You could say this is the same thing as 8 square roots of 2 over 2, which is 4 square roots of 2, plus the square root of 2 times X squared over 2.
So depending on your tastes, you might view this as more simple or this as more simple but both are equally valid.
Now I said there were multiple ways to do this.
We could have rationalized right from the get go. Let me start with our original problem.
So our original problem was 16 + 2X squared, all of that over the principle square root of 8.
We could have rationalized from the get go by multiplying the numerator and the denominator by the principle square root of 8.
And so in our denominator we'll just get 8.
And then in our numerator we would get 16 times the principle square root of 8, plus 2 times the principle square root of 8X squared.
And now we could try and simplify this a little bit more.
You can say, well, everything in the numerator and denominator is divisible by 2, so the 16 could become an 8 if you divide by 2. The 2 becomes a 1. And this 8 becomes a 4.
And then you get 8 square roots of 8 plus the square root of 8X squared. All of this over 4.
And then you say wait, this looks different from what we had over here.
And the reason is we still haven't simplified this radical.
We know that we can rewrite the principle square root of 8 as 2 square roots of 2.
And then we can see again that everything in the numerator and the denominator is also divisible by 2, so lets do that again.
So if you divide everything in the numerator by 2 you can get rid of this 2 and that 2.
And everything in the denominator by 2, this will become a 2.
So then you have 8 square roots of 2 plus the square root of 2X squared, all of that over 2.
Which is exactly what we had gotten here.