Let $a$ and $b$ be real numbers such that $a>b , 2^a +2^b=75$ and $2^{-a} + 2^{-b} =1/12$ , find the value of $2^{a-b+2}$

Question

I have tried till this how to solve further please help. $$75/12=(2^a+2^b)(2^{-a} + 2^{-b}) =2+2^{a-b}+2^{b-a}$$

Answer 1

If we multiply both equations we get $$(2^a +2^b) (2^{-a} + 2^{-b}) =75/12$$

Now let $x=2^{a-b+2}>4$. Then $$1+{x\over 4}+{4\over x}+1 = {25\over 4}$$

so $$ 17x = x^2+16\implies (x-16)(x-1)=0\implies x=16$$