Question

A forma mais simples de escrever (a^-2/b : b^-3/a^2)^-3 é:

Answer 1

Olá,

use as propriedades da exponenciação..

$\dfrac{x^m}{y^m}\cdot \dfrac{x^n}{y^n}= \dfrac{x^{m+n}}{y^{m+n}} \\\\\\ x^0=1\\\\\\ \dfrac{1}{x^1}\Longleftrightarrow x^{-1}\\\\\\ (x^m)^n=x^{mn}\\\\\ \left(\dfrac{x^{-m}}{y^{n}}\right)^{-k}=\left(\dfrac{x^m}{y^{-n}}\right)^k$

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$\left( \dfrac{a^{-2}}{b}: \dfrac{b^{-3}}{a^2}\right)^{-3}=\left( \dfrac{a^{-2}}{b}\cdot \dfrac{a^2}{b^{-3}}\right)^{-3}\\\\ \left( \dfrac{a^{-2}}{b}: \dfrac{b^{-3}}{a^2}\right)^{-3}=\left( \dfrac{a^{-2+2}}{b^{1-3}}\right)^{-3}\\\\ \left( \dfrac{a^{-2}}{b}: \dfrac{b^{-3}}{a^2}\right)^{-3}=\left( \dfrac{a^0}{b^{-2}}\right)^{-3}\\\\ \left( \dfrac{a^{-2}}{b}: \dfrac{b^{-3}}{a^2}\right)^{-3}=\left( \dfrac{1}{b^{-2}}\right)^{-3}$

$\left( \dfrac{a^{-2}}{b}: \dfrac{b^{-3}}{a^2}\right)^{-3}=\left( \dfrac{b^{2}}{1} \right)^3\\\\ \left( \dfrac{a^{-2}}{b}: \dfrac{b^{-3}}{a^2}\right)^{-3}=(b^{2})^3\\\\ \left( \dfrac{a^{-2}}{b}: \dfrac{b^{-3}}{a^2}\right)^{-3}=b^{(2)\cdot3}\\\\ \Large\boxed{\left( \dfrac{a^{-2}}{b}: \dfrac{b^{-3}}{a^2}\right)^{-3}=b^{5}}$