Question

-(5/4) elevado a -1, : por (-5/4) elevado por -4??​

Answer 1

Vamos aplicar algumas propriedade de potencias e frações.

Para facilitar, considere a legenda abaixo:

$(1):~~Potencia~de~expoente~negativo~~\left(\dfrac{a}{b}\right)^{-c}~=~\left(\dfrac{b}{a}\right)^c\\\\\\(2):~~Potencia~de~fracao~~\left(\dfrac{a}{b}\right)^{c}~=~\left(\dfrac{a^c}{b^c}\right)\\\\\\(3):~~Potencia~de~base~negativa~~(-a)^b~=~\left\{\begin{array}{cc}a^b&se~b~for~par\\-a^b&se~b~for~impar\end{array}\right$

$(4):~~Divisao~de~fracao~~\dfrac{\frac{a}{b}}{\frac{c}{d}}~=~\dfrac{a}{b}\cdot\dfrac{d}{c}\\\\\\(4):~~Divisao~de~Potencias~de~mesma~base~~\left\{\begin{array}{ccc}\dfrac{a^b}{a^c}&=&a^{b-c}\\&ou&\\\dfrac{a^b}{a^c}&=&\dfrac{1}{a^{c-b}}\end{array}\right$

Resolvendo a expressão, temos:

$-\left(\dfrac{5}{4}\right)^{-1}~:~\left(\dfrac{-5}{4}\right)^{-4}~=\\\\\\=~\dfrac{-\left(\dfrac{5}{4}\right)^{-1}}{\left(\dfrac{-5}{4}\right)^{-4}}~~~\rightarrow~~Aplicando ~a~Propriedade~(1)\\\\\\=~\dfrac{-\left(\dfrac{4}{5}\right)^{1}}{\left(\dfrac{4}{-5}\right)^{4}}~~~\rightarrow~~Aplicando~a~Propriedade~(2)\\\\\\=~\dfrac{-\dfrac{4^1}{5^1}}{\dfrac{4^4}{(-5)^4}}~~~\rightarrow~~Aplicando~a~Propriedade~(3)$

$=~\dfrac{-\dfrac{4^1}{5^1}}{\dfrac{4^4}{5^4}}~~~\rightarrow~~Aplicando~a~Propriedade~(4)\\\\\\=~-\dfrac{4^1}{5^1}~\cdot~\dfrac{5^4}{4^4}~~~\rightarrow~~Aplicando~a~Propriedade~(5)\\\\\\=~-~\dfrac{5^{4-1}}{4^{4-1}}\\\\\\=~-~\dfrac{5^3}{4^3}\\\\\\=~\boxed{-\dfrac{125}{64}}$