Question

aplique as propriedades das potências ​

Answer 1

Para esta questão, vamos utilizar basicamente 4 propriedades, como é mostrado abaixo:

1) Produto de potencias de mesma base

$b^a~.~b^c~=~b^{a+c}$

2) Divisão de potencias de mesma base

$\frac{b^a}{b^c}~=~b^{a-c}~~~~~~ou~~~~~~\frac{b^a}{b^c}~=~\frac{1}{b^{c-a}}$

3) Potencia de potencia

$\left(b^a\right)^c~=~b^{a\,.\,c}$

4) Potencia de expoente negativo

$b^{-a}~=~\frac{1}{b^\,a}$

Vamos as questões.

$a)\\\\\\7^3~.~7~.~7^2~=\\\\\\=~7^3~.~7^1~.~7^2\\\\\\=~7^{3+1+2}\\\\\\=~\boxed{7^{6}}~~~~ou~~~~\boxed{117649}\\\\\\\\b)\\\\\\2^4~.~2^{-5}~=\\\\\\=~2^{4+(-5)}\\\\\\=~2^{4-5}\\\\\\=~\boxed{2^{-1}}~~~~ou~~~~\boxed{\frac{1}{2}}$

$c)\\\\\\8^2:8~=\\\\\\=~\frac{8^2}{8^1}\\\\\\=~8^{2-1}\\\\\\=~\boxed{8^1}~~~~ou~~~~\boxed{8}$

$d)\\\\\\\left(8^3\right)^{-1}~=\\\\\\=~8^{3\,.\,(-1)}\\\\\\=~8^{-3}\\\\\\=~\boxed{\frac{1}{8^3}}~~~~ou~~~~\boxed{\frac{1}{512}}\\\\\\\\e)\\\\\\9^2~.~9:9^{-1}~=\\\\\\=~\frac{9^2~.~9^1}{9^{-1}}\\\\\\=~9^{2+1-(-1)}\\\\\\=~9^{2+1+1}\\\\\\=~\boxed{9^{4}}~~~~ou~~~~\boxed{6561}$

$f)\\\\\\\left(3^{-1}~.~3~.~3^{-3}\right)^{2}~=\\\\\\=~\left(3^{-1+1+(-3)}\right)^2\\\\\\=~\left(3^{-3}\right)^2\\\\\\=~3^{-3\,.\,2}\\\\\\=~\boxed{3^{-6}}~~~~ou~~~~\boxed{\frac{1}{3^6}}~~~~ou~~~~\boxed{\frac{1}{729}}$