# Problem of the Week Problem E Exponentially Large

## Problem

Alex can choose four different numbers $$w,x,y$$ and $$z$$ from the set $\{-1,-2,-3,-4,-5\}$ What is the largest possible value of $$w^x + y^z$$?

## Solution

Consider $$w^x$$ and choose $$w$$ and $$x$$ to be different numbers from the set $$\{-1,-2,-3,-4,-5\}$$.
What is the largest possible value for $$w^x$$?

Since $$x$$ will be negative, we write $$w^x=\dfrac{1}{w^{-x}}$$, where $$-x>0$$.

If $$x$$ is odd, then since $$w$$ is negative, and $$w^x$$ will be negative.
If $$x$$ is even, then $$w^x$$ will be positive.
So to make $$w^x$$ as large as possible, we make $$x$$ even (ie. $$-2$$ or $$-4$$).
Also, in order to make $$w^x=\dfrac{1}{w^{-x}}$$ as large as possible, we want to make the denominator, $$w^{-x}$$, as small as possible, so $$w$$ should be as small as possible in absolute value.

Therefore, the largest possible value of $$w^x$$ will be when $$w = -1$$ and $$x$$ is either $$-2$$ or $$-4$$, giving 1 in both cases (ie. $$(-1)^{-2}=(-1)^{-4}= 1$$).

What is the second largest possible value for $$w^x$$?
Again, we need $$x$$ to be even to make $$w^x$$ positive, and from above, we can assume that $$w \neq -1$$.
When $$x=-2$$, the smallest possible base (in absolute value) is $$w=-3$$ and $$w^x = \dfrac{1}{(-3)^2}=\dfrac{1}{9}$$.
When $$x=-4$$, the smallest possible base (in absolute value) is $$w=-2$$ and $$w^x = \dfrac{1}{(-2)^4}=\dfrac{1}{16}$$.
The largest of these two is $$\dfrac{1}{9}$$.

Therefore, the two largest possible values for $$w^x$$ are 1 and $$\dfrac{1}{9}$$.

Thus, looking at $$w^x + y^z$$, since $$-1$$ can only be chosen for one of these four numbers, then the largest possible value for this expression is the sum of the largest two possible values for $$w^x$$, ie. $$1 + \dfrac{1}{9}= \dfrac{10}{9}$$, which is obtained by calculating $$(-1)^{-4} + (-3)^{-2}$$.

Therefore, the largest value of $$w^x + y^z$$ is $$\dfrac{10}{9}$$. (This will occur when $$w=-1$$, $$x=-4$$, $$y=-3$$, and $$z=-2$$ or $$w=-3$$, $$x=-2$$, $$y=-1$$, and $$z=-4$$.)