Question

Com resolver: (2/5((elevado a x+3 = (125/8) elevado a x - 1 vezes (0,4) elevado a 2x-3 ??

Continuo apanhando...

Answer 1

$\left(\frac{2}{5}\right)^{x+3} = \left(\frac{125}{8}\right)^{x-1}\,\,\cdot\,\, 0,\!4^{2x-3}$

Observe que :

$\frac{125}{8} = ( \frac{5}{2})^{3} \\ \\ 0,\!4= \frac{4}{10} = \frac{2}{5}$

Então:

$\left(\frac{2}{5}\right)^{x+3}= \left(\frac{5}{2}\right)^{3(x-1)}\,\,\, \cdot \left(\frac{2}{5}\right)^{2x-3} \\ \\ \left(\frac{2}{5}\right)^{x+3}= \left(\frac{5}{2}\right)^{3x-3}\,\,\, \cdot \left(\frac{2}{5}\right)^{2x-3}$

Agora, observe que:

$\frac{5}{2} = \left( \frac{2}{5}\right)^{-1}$

Então:

$\left(\frac{2}{5}\right)^{x+3}= \left(\frac{2}{5}\right)^{(-1)\cdot3x-3}\,\,\, \cdot \left(\frac{2}{5}\right)^{2x-3} \\ \\ \left(\frac{2}{5}\right)^{x+3}= \left(\frac{2}{5}\right)^{3-3x}\,\,\, \cdot \,\,\,\left(\frac{2}{5}\right)^{2x-3}$

Agora, lembre-se que:

Multiplicação de potencias de mesma base: conserve a base, some os expoentes:

$\left(\frac{2}{5}\right)^{x+3}= \left(\frac{2}{5}\right)^{3-3x+2x-3} \\ \\ \left(\frac{2}{5}\right)^{x+3}= \left(\frac{2}{5}\right)^{-x}$

Bases iguais, expoentes iguais

$x+3=-x \\ x+x=-3 \\ 2x=-3 \\ \\ \boxed{x=- \frac{3}{2} }$