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Problem of the Week
Problem D and Solution
The Same Power


Sometimes two powers that are not written with the same base are still equal in value. For example, \(9^3 = 27^2\) and \((-5)^4=25^2\).

If \(x\) and \(y\) are integers, find all ordered pairs \((x,y)\) that satisfy the equation \[(x-1)^{x+y}=8^2\]


Since \(8^2 = 64\), we want to look at how we can express \(64\) as \(a^b\) where \(a\) and \(b\) are integers. There are six ways to do so. We can do so as \(64^1\), \(8^2\), \(4^3\), \(2^6\), \((-2)^6\), and \((-8)^2\).

We use these powers and the expression \((x-1)^{x+y}\) to find values for \(x\) and \(y\).

Therefore, there are six ordered pairs that satisfy the equation.
They are \((65,-64)\), \((9,-7)\), \((5,-2)\), \((3,3)\), \((-1,7)\), and \((-7,9)\).