A Big Number

unit digit in 22222^55555 is same as 2^55555 unit digit in this = 2^3 = 8 (to find this index/4 find remainder 55555/4 remainder is 3 so unit digit is 2^3 = 8

unit digit in 55555^22222 is same as 5^2 unit digit in this = 5

(8 + 5)^33333 unit digit is 3

similarly unit digit in (33333^77777 + 77777^33333)^44444 = 0^44444 = 0

answer is 3 + 0 = 3

${ ({ 22222 }^{ 55555 }+{ 55555 }^{ 22222 }) }^{ 33333 }+{ ({ 33333 }^{ 77777 }+{ 77777 }^{ 33333 }) }^{ 44444 }$

First, to analyse, I have that: ${ 22222 }^{ 55555 }\equiv { 2 }^{ 55555 }\equiv { 2 }^{ 5(11111) }\equiv { 2 }^{ 11111 }\equiv 8\quad (mod\quad 10)$

${ 55555 }^{ 22222 }\equiv { 5 }^{ 22222 }\equiv 5^{ 2(11111) }\equiv 5^{ 11111 }\equiv 5\quad (mod\quad 10)$

So $8+5=3 and { 3 }^{ 33333 }\equiv 3\quad (mod\quad 10)$ .

$\\ { 33333 }^{ 77777 }\equiv { 3 }^{ 77777 }\equiv 3^{ 7(11111) }\equiv 7^{ 11111 }\equiv 3\quad (mod\quad 10)$

${ 77777 }^{ 33333 }\equiv { 7 }^{ 33333 }\equiv 7^{ 3(11111) }\equiv 7^{ 11111 }\equiv 7\quad (mod\quad 10)$

So ${ (3+{ 7 }) }^{ 44444 }={ (0) }^{ 44444 }\equiv 0\quad (mod\quad 10)$

Them the answer is 3.