Question

Daca $4^{x} +4^{-x} =23$  atunci cat este  $2^{x} + 2^{-x}$

Answer 1

$4^x + 4^{-x} = 23 \Rightarrow 2^{2x}+ 2^{-2x} = 23 \Rightarrow \big(2^x+2^{-x}\big)^2-2\cdot 2^x\cdot 2^{-x} = 23 \Rightarrow \\ \Rightarrow \big(2^x+2^{-x}\big)^2-2\cdot 2^{x-x} = 23 \Rightarrow \big(2^x+2^{-x}\big)^2 - 2\cdot 2^0 = 23 \Rightarrow \\ \Rightarrow \big(2^x+2^{-x}\big)^2 - 2 = 23 \Rightarrow \big(2^x+2^{-x}\big)^2 = 23+2 \Rightarrow \\ \Rightarrow \big(2^x+2^{-x}\big)^2= 25 \Rightarrow 2^x+2^{-x} = \pm \sqrt{25}$

$(Dar, functiile exponentiale sunt intotdeauna pozitive)$

$\Rightarrow 2^x+2^{-x} = + \sqrt{25} \Rightarrow \boxed{2^x+2^{-x} = 5}$