How to simplify this equation? $1+\sqrt {2^{2a_{n}+b_{n}+1}-16^{a_{n}}-4^{b_{n}}}=\log _{3}\left( a_{n}+b_{n}\right) $

Question

How to simplify this equation? $1+\sqrt {2^{2a_{n}+b_{n}+1}-16^{a_{n}}-4^{b_{n}}}=\log _{3}\left( a_{n}+b_{n}\right) $

Answer 1

Assume $a=a_n$ & $b=b_n$.

$$2^{2a+b+1}-16^a-4^b=-16^a+2^{2a+b+1}-4^b$$

$$=-(16^{a}-2\times2^{2a}\times2^b+4^b)$$

$$=-\left((4^{2a})-2\times (4^a)\times(2^b)+(2^{2b})\right)$$

Note the similarity of this to the expansion of $(x-y)^2$. $$-\left((4^{2a})-2\times (4^a)\times(2^b)+(2^{2b})\right)=-\left((4^a)-(2^b)\right)^2$$

I think you can take it from here. Sorry if there's any errors or if I've been unclear.