Back To Basics II

Algebra Level 5

\(x,y,\) and \(z\) are real numbers satisfying the equation \(x^{2}+y^{2}+z^{2}=1.\)

The maximum value of \(3xy^{2}z\) can be expressed in the form \(\dfrac{a}{b}\) for co-prime positive integers \(a\) and \(b\).

Find \(a+b\).


Can you generalize the maximum value of \(x^{m}y^{n}z^{p}\) for positive integers \(m,n,p\) given the same constraints?


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