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Simplify x y to the -2 power, times 2 x-squared over y-squared to the 4th power

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So let's see what we can do over here

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So let's think about this part of our expression first, if I have x over y to the n-th power

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we know that this is the same thing as, x to the n over y to the n

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So let's apply that right over here

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So this expression, 2 x-squared over y-squared to the 4th

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this is going to be the same as, 2 x-squared to the 4th power, over y-squared to the 4th power

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and of course we still have our x y to the -2 power out here, and actually let me simplify that as well

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we know that if we have, we know that if we have, x times y to the n-th power

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this the same thing as, taking each of the powers of each of the x and the y, and then taking the product

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this is x to the n times y to the n

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and actually if you just expand, if you multiply xy times itself n times

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we will see why this actually works out

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because you will have x multiplied by itself n times, if you rearrange the product

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and y multiplied by itself n times

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so if we use the same property here, x y to the -2, that's going to the the same exact thing as

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x to the -2 times y to the -2

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now, what can we simplify?

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well, we can use this property again, right over here, we have 2 x-squared raised to the 4th power

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so that's going to be the same as, that's going to the the same thing as

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2 to the 4th power, 2 to the 4th power times x-squared to the 4th power

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now we can do it in the denominator, y-squared to the 4th power

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Actually are going to use a different property in the denominator

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We just have y-squared to the 4th power

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We know that if we have x to the n, and then raise that to the m-th power

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this is the same as, x to the n times m

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so y-squared, and then raising that to the 4th power, that's going to be equal to

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y to the 2 times 4 power, or y to the 8th power.

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And of course out here, we have x to the -2 times y to the -2

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Now let's think a little bit about what x to the -2 and y to the -2 mean.

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x to the -2 just means 1 over x-squared, y to the -2 just means 1 over y-squared

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so let's rewrite it like that, this is the same thing as, 1 over x-squared times 1 over y-squared

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So we can just write that as 1 over x-squared, y-squared. That and that are completely equivalently

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So let's see what we can do now, so we have one last thing. Well we don't have one last thing

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we have a few more things to worry about

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we have x-squared and then we are raising that to the 4th power

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once again, we can use this property right here, this is going to be x to the 2 times 4 power

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so this part right over there, do it in a new color, this part right over here is going to simplify to

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x to the 8th power, 2 times 4

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this part right over here is going to, 2 to the 4th power, 2 to the 3rd is 8. So 2 to the 4th is 16

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so that is 16. and so our whole expression has now simplified to, if we multiply it times this

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we multiply it times 1, so i am going to multiply these 2 fractions.

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My numerator is 16 times x to the 8th, 16 times x to the 8th, and all of that over

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I have x-squared, I have an x-squared. And then I am going to have a y-squared times a y to the 8th

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what is y-squared times y to the 8th? y-squared times y to the 8th

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we have the same base, so we can add the exponents for taking the products

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so it's going to be y to the 2 plus 8, or y to the 10th power

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so y-squared times y to the 8th is going to be y to the 10th power

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and we are almost there, we have one more thing we can simplify

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we have x to the 8th over x-squared, and we know, I will just write the general property

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x to the m over, over, x to the n, this is equal to x to the m minus n power

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so x to the 8th over x-squared, that over there will simplify to x to the 8 minus 2 power

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or x to the 6th power

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and our whole expression has now simplified to 16, 16 times x to the 6th, times x to the 6th

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all of that over y to the 10th, and we are done