Question

a^b×b^a+a^b+b^a=5329. then a^b=?​

Answer 1

We're given,

$a^bb^a+a^b+b^a=5329$

Add 1 to both sides.

$a^bb^a+a^b+b^a+1=5330$

Now we factorize both sides as the following.

$(a^b+1)(b^a+1)=2\times 5\times 13\times 41$

By checking some possible combinations in the RHS, I get that,

$(a^b+1)(b^a+1)=65\times 82=(64+1)(81+1)=(4^3+1)(3^4+1)$

Hence  $a^b=64\quad\text{OR}\quad a^b=81.$