Question

Se o quociente de $64^{x-1}$ por $4^{x-1}$ é $256^{2x}$ , então x é ?

Answer 1

$\frac{64^{x-1}}{4^{x-1}}=4^{8x}\\\\\frac{4^{3x-3}}{4^{x-1}}=4^{8x}\\4^{(3x-3)-(x-1)}=4^{8x}\\4^{2x-2}=4^{8x}\\2x-2=8x\\2x-2=8x\\-6x=2$

$\boxed{x=\frac{-1}{3}}$

Answer 2

$64^{x-1}=(2^6)^{x-1}=2^{6x-6}$

$4^{x-1}=(2^2)^{x-1}=2^{2x-2}$

$256^{2x}=(2^8}^{2x}=2^{16x}$

Assim

$\dfrac{2^{6x-6}}{2^{2x-2}}=2^{16x}$

Veja que, $\dfrac{2^{6x-6}}{2^{2x-2}}=2^{6x-6-2x+2}=2^{4x-4}$.

Logo

$2^{4x-4}=2^{16x}$

$4x-4=16x$

$16x-4x=-4$

$12x=-4$

$x=\dfrac{-4}{12}$

$x=\dfrac{-1}{3}$