my question is the same as the title. I am trying to learn the rules of exponentiation. So I am trying to solve \displaystyle xy\cdot \frac{x}{y} = ?

Hey, goofygoober123, let’s go through this step by step:

We can write \displaystyle \frac{x}{y} as xy^{-1}

So now we’re working with (xy)\cdot (xy^{-1}), but because of the associative property for multiplication, we can rearrange the parentheses to include the whole product, like (xyxy^{-1}). Now, use the associative property *again* to move the x's together and the y's together, like so:

xyxy^{-1} = xxyy^{-1} = (xx)\cdot (yy^{-1}),

and then add the exponents together:

(xx)\cdot (yy^{-1}) = x^{1+1}\cdot y^{1-1} = x^2 \cdot y^{0} = x^2 \cdot 1 = x^2,

and that’s the final solution! I hope my attention to detail helps you see what’s exactly going on.

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