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Simplify 3 times the square root of 200x to the third.
So a good place to start, let's factor out 200.
So if I'm just focusing on the 200 here, 200 is 2 times 100.
And 100 is a perfect square.
It's 10 times 10.
And then if we were to factor x to the third, that's the
same thing as x squared times x.
So let's write that out.
So let me just rewrite.
This whole thing is equal to 3 times the square root-- I'll
write it all under one big radical sign right now-- of 2
times 100 times x squared times x.
Stretch the radical sign.
So the x squared times x, that is x to the third.
And then the 200 is 2 times 100.
Now, we can you think of this as all of these numbers, the
product of all these numbers to the 1/2 power or taking the
square root of the product, that's like taking the square
root of-- or that's equivalent to-- taking the square root of
each of these terms. So this is going to be equal to 3
times the square root of 2 times the square root of 100
times the square root of x squared times the
square root of x.
These terms right there in blue, that's
these terms right there.
The terms in orange, that's those terms right over there.
Now, several of these are not too hard to evaluate.
So we have our 3 out front.
So this is going to be equal to 3.
And then let me mark off the terms. Let's do this one the
first.
What's the square root of 100?
Or the principal root of 100?
The positive square root.
Well it's 10.
10 times 10 is 100.
It's going to be 3 times 10.
That's that term right there.
And then, what is the square root of x squared?
Well it's just x. x times x is x squared.
So times x.
And then we have to deal-- so we've dealt with that, we've
dealt with that, we've dealt with that.
And then we have these two terms right here where we
can't take the square roots.
So we could say times the square root of 2 times the
square root of x.
So this term right here times the square root of 2.
And then, finally, out here times the square root of x.
So if we want to simplify this fully, we'd get 3 times 10
times x, which is 30x.
And then we could actually merge these.
The square root of 2 times the square root of x.
That's the same thing as the square root of 2x.
So we could say times the square root of 2x.
And we have simplified this as far as it can go.