Question

RESPONDAM POR FAVOR ;_;

Determine o valor de cada uma das expressões:
$\frac{ { - 3}^{ - 3} + {2}^{ - 2} }{ {2}^{ - 3} } \\ \\ {3}^{ - 1} + {3}^{ - 2} - ( - 3)^{ - 3}$

Answer 1

Olá.

Propriedades usadas:

$\mathsf{a^{-n}=\dfrac{1}{a^n}}$

     Primeira questão

$\mathsf{\dfrac{{-3}^{-3}+{2}^{-2}}{{2}^{-3}}=}\\\\\\ \mathsf{\left({-3}^{-3}+{2}^{-2}\right)\div{2}^{-3}=}\\\\\\ \mathsf{\left(-\dfrac{1}{3^3}+\dfrac{1}{2^2}\right)\div\dfrac{1}{2^3}=}\\\\\\ \mathsf{\left(-\dfrac{1}{27}+\dfrac{1}{4}\right)\div\dfrac{1}{8}}$

Para igualar os denominadores, multiplicarei ambas as frações por uma outra nula (que no final vale 1), que tem no numerador e denominador o denominador da outra fração. Teremos:

$\mathsf{\left(-\dfrac{1}{27}+\dfrac{1}{4}\right)\div\dfrac{1}{8}=}\\\\\\ \mathsf{\left(-\dfrac{1}{27}\cdot\dfrac{4}{4}+\dfrac{1}{4}\cdot\dfrac{27}{27}\right)\div\dfrac{1}{8}=}\\\\\\ \mathsf{\left(-\dfrac{4}{108}+\dfrac{27}{108}\right)\div\dfrac{1}{8}=}\\\\\\ \mathsf{\left(\dfrac{-4+27}{108}\right)\div\dfrac{1}{8}=}\\\\\\ \mathsf{\left(\dfrac{23}{108}\right)\div\dfrac{1}{8}=}\\\\\\ \mathsf{\left(\dfrac{23}{108}\right)\cdot\dfrac{8}{1}=}\\\\\\ \mathsf{\dfrac{23\cdot8}{108}=}\\\\\\ \mathsf{\dfrac{184}{108}}$

Essa fração pode ser reduzida, a partir do número 4. Teremos:

$\mathsf{-\dfrac{184}{108}\rightarrow-\dfrac{184^{:4}}{108^{:4}}=-\dfrac{46}{27}}$

     Segunda questão

$\mathsf{{3}^{-1}+{3}^{-2}-(-3)^{-3}=}\\\\\\ \mathsf{\dfrac{1}{3}+\dfrac{1}{3^2}-\left(-\dfrac{1}{3^3}\right)=}\\\\\\ \mathsf{\dfrac{1}{3}\cdot\dfrac{3^2}{3^2}+\dfrac{1}{3^2}\cdot\dfrac{3}{3}+\dfrac{1}{3^3}=}\\\\\\ \mathsf{\dfrac{3^2}{3^3}+\dfrac{3}{3^3}+\dfrac{1}{3^3}=}\\\\\\ \mathsf{\dfrac{3^2+3+1}{3^3}=}\\\\\\ \mathsf{\dfrac{9+3+1}{27}=}\\\\\\ \boxed{\mathsf{\dfrac{13}{27}}}$

Qualquer dúvida, deixe nos comentários.

Bons estudos$.$