The Magnificent Operator

Let me define an operator for two natural numbers \(a,b\) as \([[]a,b[]]\)

\([[]a,b[]]=\frac{a^b}{b^a}\frac{b^b}{a^a}\)

Now, if \(a+b\leq15\), with \(b\geq8\), find the number of ordered pairs \((a,b)\) for which \([[]a,b[]]\) is a perfect square of an integer.

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