If $a^b = b^a$ and $a=2b$ then find the value of $a^2+b^2$ If $a^b = b^a$ and $a=2b$ then find the value of $a^2+b^2$
My Attempt, 
$$a^b = b^a$$
$$a^b =b^{2b}$$
$$a^b =b^b.b^b$$.
Now, what should I do further? 

We know that $a=2b $, so we have $$a^b = b^b \cdot b^b \Rightarrow (2b)^b = b^b \cdot b^b \Rightarrow 2^b = b^b $$. Assuming $b\in \mathbb Z $, we get, $b=2$ and thus $a=4$, so thus $\boxed {a^2+b^2=20} $. Hope it helps. 
