# A hardly known factorization

If you get the command to factorize *a*^{4} + *b*^{4} most people with a little sense of mathematics tell you

that it cannot be done because the sum of two squares cannot be factorized and

*a*^{4} and *b*^{4} are a square each for sure.

*a*^{4} = (*a*^{2})^{2}, so a square.

It really can be factorized. Look.

*a*^{4} + *b*^{4} =

*a*^{4} + 2(*ab*)^{2} + *b*^{4} - 2(*ab*)^{2} =

(*a*^{2} + *b*^{2})^{2} - (*ab*√2)^{2} =

(*a*^{2} - *ab*√2 + *b*^{2})(*a*^{2} + *ab*√2 + *b*^{2})

On this site is not discussed, whether the found two factors can be factorized.

## Higher exponents of *a* and *b*

As well is right: *a*^{4n} + *b*^{4n} = (*a*^{2n} - (*ab*)^{n} √2 + *b*^{2n})(*a*^{2n} + (*ab*)^{n} √2 + *b*^{2n})

The *n* is a natural number.