# Problem of the Week Problem D and Solution The Same Power

## Problem

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 $(x-1)^{x+y}=8^2$

## Solution

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 $$y$$.

• The power $$(x-1)^{x+y}$$ is expressed as $$64^1$$ when $$x-1 = 64$$ and $$x+y = 1$$. Then $$x=65$$ and $$y=-64$$ follows. Thus $$(65,-64)$$ is one pair.

• The power $$(x-1)^{x+y}$$ is expressed as $$8^2$$ when $$x-1 = 8$$ and $$x+y=2$$. Then $$x=9$$ and $$y=-7$$ follows. Thus $$(9,-7)$$ is one pair.

• The power $$(x-1)^{x+y}$$ is expressed as $$4^3$$ when $$x-1 = 4$$ and $$x+y = 3$$. Then $$x=5$$ and $$y=-2$$ follows. Thus $$(5,-2)$$ is one pair.

• The power $$(x-1)^{x+y}$$ is expressed as $$2^6$$ when $$x-1 = 2$$ and $$x+y = 6$$. Then $$x=3$$ and $$y=3$$ follows. Thus $$(3,3)$$ is one pair.

• The power $$(x-1)^{x+y}$$ is expressed as $$(-2)^6$$ when $$x-1 = -2$$ and $$x+y = 6$$. Then $$x=-1$$ and $$y=7$$ follows. Thus $$(-1,7)$$ is one pair.

• The power $$(x-1)^{x+y}$$ is expressed as $$(-8)^2$$ when $$x-1 = -8$$ and $$x+y = 2$$. Then $$x=-7$$ and $$y=9$$ follows. Thus $$(-7,9)$$ is one pair.

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