Question

$<var>(10^{x})^{x-1} = \frac{1}{10^{6}}</var>$

Answer 1

Lembrando algumas propriedades de potenciação:

$(a^{b})^{c} = a^{bc}$
$\frac{1}{a^{b}} = a^{-b}$

E lembrando que: $a^{b} = a^{c} <=> b = c$
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$(10^{x})^{x-1} = \frac{1}{10^{6}}$
$10^{x(x - 1)} = 10^{-6}$
$10^{x^{2} - x} = 10^{- 6}$

Bases iguais, iguale os expoentes:

$x^{2} - x = - 6$
$x^{2} - x + 6 =0$

a = 1
b = - 1
c = 6

$D = b^{2} - 4ac$
$D = (-1)^{2} - 4*1*6$
$D = 1 - 24$
$D = - 23$
$\sqrt{D} = \sqrt{- 23}$
$\sqrt{D} = \sqrt{23} * \sqrt{- 1}$
$\sqrt{D} = \sqrt{23} * i$
$\sqrt{D} = i\sqrt{23}$

$x = \frac{- b +- \sqrt{D} }{2a}$

$x = \frac{-(-1) +- i\sqrt{23} }{2*1}$

$x = \frac{1 +- i\sqrt{23} }{2}$

$x' = \frac{1 + i\sqrt{23} }{2}$

$x'' = \frac{1 - i\sqrt{23} }{2}$