Fermat helps: $\sqrt [ 52 ]{ { 473 }^{ 156 }+{ 981 }^{ 13 } } =105823817$

Consider $\large (473^{12})^{13} + 981^{13} \neq (105823817^4)^{13}$

$\large \Rightarrow \sqrt [ 52 ]{ { 473 }^{ 156 }+{ 981 }^{ 13 } } \neq 105823817$

Fermat's last theorem says, that no $x^{n} + y^{n} = z^{n}$ has no positive integral (integer) solutions for $n > 2$ .

Rearranging:

$473^{156}+981^{13}=105823817^{52}$

Rewriting:

$(473^{12})^{13}+981^{13}=(105823817^{4})^{13}$

Due to Fermat's theorem, it must have no solutions ( $n = 13$ ).

Checking with Calculator (don't do this, just trying to show)

$473^{156} + 981^{13} =$

1898182455987008946396006363347308090945891508259974310597456828126172089112739214102767043783335566247400837579465612442872991162298052157315387100663665430703590466947887678679235884216295902469587608194039586910361116996504612190227718265817732730585408218274194030911655001269442358944233592451831877370746586205453731339911485478915302078897214317241051804145723391462653591708006051388370018769028511822619781702

$105823817^{52} =$

1898182455987008946396006363347308090945891508259974310597456828126172089112739214102767043783335566247400837579465612442872991162298052157315387100663665430703590466947887678679235884216295902469587608194039586910361116996504612190227718265817732730585408218274194030911655001269442358944233592451831877370746586205453731339911485478915302078897214317241051804145723391462653590928719653394306258535732918833303945761

Your calculator could round this and call it equal, even though it is not.