Question

Como simplificar a expressão
(2a²)⁴ . (4a³)³.(b⁵)
_____________
(32a)². (b⁵)³

(Fatoração)

Answer 1

Fiz passo a passo para que pudesse entender melhor. Apliquei algumas propriedades como:

$(a.b)^{2}=a^{2}.b^{2}\\ a^{2}.a^{3}=a^{2+3}=a^{5}\\ (a^{2})^{3}=a^{2.3}=a^{6}\\ \frac{a^{2} }{a^{3} } =a^{2-3}=a^{-1}= \frac{1}{a}$


$\frac{(2a^{2})^{4}.(4a^{3})^{3}.(b^{5})}{(32a)^{2}.(b^{5})^{3}}=\\ \frac{(2.a^{2})^{4}.(2^{2}.a^{3})^{3}.b^{5}}{(2^{5}.a)^{2}.(b^{5})^{3}}=\\ \frac{(2^{4}.a^{2.4}).(2^{2.3}.a^{3.3}).b^{5}}{(2^{5.2}.a^{2}).(b^{5})^{3}}=\\ \frac{2^{4}.2^{6}.a^{8}.a^{9}.b^{5}}{2^{10}.a^{2}.b^{15}}=\\ \frac{2^{4+6}.a^{8+9}.b^{5}}{2^{10}.a^{2}.b^{15}}=\\ \frac{2^{10}.a^{17}.b^{5}}{2^{10}.a^{2}.b^{15}}=\\ \frac{a^{17}.b^{5}}{a^{2}.b^{15}}=\\ a^{17-2}.b^{5-15}}=a^{15}.b^{-10}\\= \frac{a^{15}}{b^{10}}$