x in the power of 4 -x +1=0 ?????

This might mean x⁴ - x + 1 = 0

However, there are no real solutions to this.  There are 4 complex solutions given (approximately) by:

0.7271 - 0.43i

-0.7271 - 0.9341i

0.7271 + 0.43i

-0.7271 + 0.9341i

x=5

It really does, it equals 5

But there are no powers here.  

x in the power of 4 -x +1=0 ?????

Well, you can restate it. 

4-^x+1=0

but the equation will not become true. 

So, Melody is a big time champion!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

So.....I assume we have

x⁴ - x + 1 = 0     ???

There's not much we can do here, algebraically...let's look at the graph here.....https://www.desmos.com/calculator/u63grqdwio

Note that this function never intersects the x axis so we have no real roots or solutions !!!

(I hope this is what you have asked.....)

you think 

$$\\4^{-x}+1=0\\ 4^{-x}=-1\\ \frac{1}{4^x}=-1\\ 4^x=-1\\ 4^x \mbox{ cannot be negative so there is no real solution (Thanks for reminding me Alan.)}$$

what graph Chris?