Raise any integer to any integer power: to find all integers that have integer roots for any integer power

*n*,*c*

^{n}

and

^{n}√(*c*^{n}) =*c*. Let*c*=*a*+*b*then*c*

^{n}= (

*a*+

*b*)

^{n}

and it follows

(

*a*+*b*)^{n}
also finds

**all**integers (from integer binomial pairs of a and b) having an integer root for a power n (as for*c*^{n}).
Given the above equality, then the expression

*a*

^{n}+ (

*a*+

*b*)

^{n}

equates to an integer that

**does not have an****integer root**of the integer power n*n*> 2 (where the*n*= 2 case is demonstrated here).
Equivalently

(

equates to an integer that*a*+*b*)^{n}−*a*^{n}**does not have an****integer root**of the integer power*n*> 2. For the case*n*= 2 this expression is the*difference of two squares*identity.
For all positive integers a, b and n, ((

*a*+*b*)^{n})^{(1)/(n)}∈ ℤ It follows that for n greater than two (*a*^{n}+ (*a*+*b*)^{n})^{(1)/(n)}∉ℤ.