Question

$( 1 - \frac{2}{3} x) {}^{6}$

a)8/9
b)-8/9
c)8/64
d)32/41
e)-64/81
Me ajudem!!
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Answer 1

Resposta:

$\boxed{\mathtt{E}}$

Explicação passo-a-passo:

Por Binômio de Newton, temos:

$\\ \displaystyle \mathsf{\left ( 1 - \frac{2}{3}x \right )^6 = \binom{6}{0} \cdot (1)^6 \cdot \left ( - 2/3 \right )^0 + \binom{6}{1} \cdot (1)^5 \cdot \left ( - 2/3 \right )^1 + \hdots + \binom{6}{6} \cdot (1)^0 \cdot \left ( - 2/3 \right )^6}$

Portanto, o termo de quinto grau é dado por $\displaystyle \mathtt{\binom{6}{5} \cdot (1)^1 \cdot \left ( - 2/3 \right )^5}$.

Desenvolvendo,

$\\ \displaystyle \mathsf{\binom{6}{5} \cdot (1)^1 \cdot \left ( - 2/3 \right )^5 =} \\\\\\ \mathsf{\frac{6!}{5!(6 - 5)!} \cdot 1 \cdot \frac{- 32}{3 \cdot 81} =} \\\\\\ \mathsf{\frac{6 \cdot 5!}{5!1!} \cdot \frac{- 32}{3 \cdot 81} =} \\\\\\ \mathsf{\frac{- 2 \cdot 32}{81} =} \\\\\\ \boxed{\boxed{\mathsf{- \frac{64}{81}}}}$