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We're asked to divide and simplify.

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And we have one radical expression over another radical expression. The key

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to simplify this is to realize I have the principal square root of x

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over the principal root of y, this is the same thing

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as the principal root of x over y.

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And this really comes out of the exponent properties. If I have two things,

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that I take to the some power, and taking the principal root is the same thing as taking it to the

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one-half power. If I'm raising them to some power, and then

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dividing, that's the same thing as dividing first, and then raising them

10
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to that power. So let's apply that over here.

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This expression over here is going to be the same thing as the principal root,

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the principal - it's hards to write a radical sign that big - the principal root of

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sixty x-squared y over forty-eight x.

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And then we can first look at the coefficient terms, or the coefficients of, each of

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these expressions, and try to simplify that. Both the numerator and

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and the denominator is divisible by twelve, sixty divided by

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twelve is five, forty-eight divided by twelve is four.

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Both the numerator and denominator is divisible by x.

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X-squared, divided by x is just x.

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X divided by x is 1. Anything we divide the numerator by, we have to divide the

21
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denominator by, and that's all we have left. So, if we want

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to simplify this, this becomes, this is equal to, -the- -the-

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Make the radical sign.

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And then we have five-fourth's,

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five-fourth's

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And actually we can write it in a slightly different way, but I'll write it this way, five-fourth's, and we have

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nothing left in the denominator other than that four, and in the numerator we have an x, and we have

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a y. We have an x and we have a y.

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And, now, we could it just like that,

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but we might want to take more things out of the radical sign. And so one possibility

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that you can do, is that you can say that this is really the same thing as

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this is equal one-fourth times five, times five xy,

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all of that under the radical sign,

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and this is the same thing as the square root of, or the principal square root of,

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one-fourth times the principal root of

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five xy, and the square root of one-fourth. If you think about it, that's just

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one-half times one-half. Or another way you could think about it is, this is the same thing,

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this right here is the same thing as -

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So you could just say that this is one-half, one-half times one-half is one-fourth.

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Or if you don't realize it's one-half, you say, hey, this is the same thing as the square root of one over

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the square root of four, and the square root of one is one,

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and the square root - the principal root of four is two, so you

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get one-half once again. And so if you simplify this right here to one-half,

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then the whole thing can simplify to one-half times

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the principal root.

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I'll just write it all in a orange, times the principal root of

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five xy, and there's nothing else that you can really take out of the radical time here.

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Nothing else here is a perfect square.