Around the corner

Calculus Level pending

You wish to retrieve an object placed on the top of a high shelf in your living room, that is only accessible by using a ladder. To reach the living room, you have to walk down two passages, of widths a and b, which meet at a right-angled corner. The longest ladder that can be carried horizontally around this corner can be written in the form : \[(a^{x}+b^y)^z, \qquad x,y,z \in \mathbb Q\]

What is \((x+y)*z\)?

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