Question

Determine a soma das raízes da equação:

$\frac{16^x + 64}{5} = 4^{x+1}$

Answer 1

E aeee mano,

use as propriedades da exponenciação..

$\boxed{(a^m)^n~\to~a^{m\cdot n}~\to~a^{mn}}\\\\ \boxed{a^{m+n}~\to~a^m\cdot a^n}$

$\dfrac{16^x+64}{5}=4^{x+1}\\\\ (4^2)^x+64=5\cdot4^x\cdot4^1\\ (4^x)^2+64=20\cdot4^x\\ (4^x)^2-20\cdot4^x+64=0\\\\ \Delta=b^2-4ac\\ \Delta=(-20)^2-4\cdot1\cdot64\\ \Delta=400-256\\ \Delta=144\\\\ 4^x=\dfrac{-(-20)\pm \sqrt{144} }{2\cdot1}= \dfrac{20\pm12}{2}\begin{cases}4^{x'}= \dfrac{20-12}{2}= \dfrac{8}{2}=4\\\\ 4^{x''}= \dfrac{20+12}{2}= \dfrac{32}{2}=16\end{cases}$

Então..

$4^{x'}=4~~~~~~~~~~~~~~~4^{x''}=16\\ 4^{x'}=4^1~~~~~~~~~~~~~~4^{x''}=4^2\\ \not4^{x'}=\not4^1~~~~~~~~~~~~\not4^{x''}=\not4^2\\\\ x'=1~~~~~~~~~~~~~~~~~x''=2\\\\ A~soma~das~raizes...\\\\ x'+x''=1+2\\\\ \huge\boxed{\boxed{x'+x''=3}}$

Tenha ótimos estudos ;D