[tex]4^a+4^{-a} = 4 \\ \\ 16^a+16^{-a} = (4^2)^a+(4^2)^{-a} = (4^a)^2+(4^{-a})^2 \\ \\ \text{Notam }4^a = x \Rightarrow 4^{-a} = \dfrac{1}{x} \\ \\ x+\dfrac{1}{x} = 4 \Big|^2 \Rightarrow x^2+2\cdot x\cdot \dfrac{1}{x}+\Big(\dfrac{1}{x}\Big)^2 = 16 \\ \\ \Rightarrow x^2+2+\Big(\dfrac{1}{x}\Big)^2 = 16 \Rightarrow x^2+\Big(\dfrac{1}{x}\Big)^2 = 14 \Rightarrow \\ \\ \Rightarrow (4^a)^{2}+(4^{-a})^2 = 14\Rightarrow\\ \\ \Rightarrow \boxed{16^a+16^{-a} = 14}[/tex]
