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One way to think about what's going on in this equation is that we multiply both sides of the original equation
1) z = (x^1/2) / (y^1/2)
by z. That gives us
2) z^2 = z * (x^1/2) / (y^1/2)
Now, we already know from statement (1) that z = (x^1/2) / (y^1/2), so we can rewrite the right side of the equation as follows:
3) z^2 = [ (x^1/2) / (y^1/2) ] * [ (x^1/2) / (y^1/2) ]
which simplifies to
4) z^2 = x^2 / y^2
Because the left and right side of the equation are equal, multiplying anything by 'z' will give the same result as multiplying by (x^1/2)/(y^1/2)
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Your instinct is right when you say that multiplying by a constant has a different kind of effect than multiplying by a variable. For example, the equation
z = xy
returns the following when both sides are multiplied by 2:
2z = 2xy
This is relatively straightforward. But when we SQUARE both sides, what we're really doing is what I guided us through above:
1) z = xy
2a) z*z = xy * z
2b) z^2 = xyz AND z=xy [from statement 1]
3a) z^2 = xy(xy)
3b) z^2 = x^2y^2
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So that's it. A little tricky to explain with ASCII, but hopefully that helps!
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