Question

Sabendo que x = (5²)³ . (5³ : 5²) e y = (5⁹)² : (5⁴ . 5²)², qual é a potência de 5 que representa o valor de x : y?
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Answer 1

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Propriedades das potências

$\blue{\Large\boxed{\begin{array}{l}\boxed{\tt 1}~~~\sf a^m\cdot a^n=a^{m+n}\\\boxed{\tt 2}~~~\sf a^m\div a^n=a^{m-n}\\\boxed{\tt 3}~~~\sf (a^m)^n=a^{m\cdot n}\\\boxed{\tt 4}~~~\sf(a\cdot b)^n=a^n\cdot b^n\\\boxed{\tt 5}~~~\sf\bigg(\dfrac{a}{b}\bigg)^n=\dfrac{a^n}{b^n}\end{array}}}$

$\boxed{\begin{array}{l}\underline{\rm c\acute alculos~auxiliares\!:}\\\sf x=(5^2)^3\cdot(5^3\div5^2)\\\sf x=5^{2\cdot3}\cdot5^{3-2}\\\sf x=5^6\cdot 5\\\sf x=5^{6+1}\\\sf x=5^7\\\sf y=(5^9)^2\div(5^4\cdot 5^2)^2\\\sf y=5^{9\cdot2}\div(5^{4+2})^2\\\sf y=5^{18}\div(5^6)^2\\\sf y=5^{18}\div5^{12}\\\sf y=5^{18-12}\\\sf y=5^6\end{array}}$

$\Large\boxed{\begin{array}{l}\sf \dfrac{x}{y}=\dfrac{5^7}{5^6}\\\\\sf\dfrac{x}{y}=5^{7-6}\\\\\sf\dfrac{x}{y}=5\end{array}}$