Question

1) DENTIFIQUE OS NUMEROS RACIONAIS COM(Q) E OS NÚMEROS IRRACIONAIS (1)
A) ( ) 2,354329...
B)( ) 3.141414...
c) ( ) 5,324324324...
D) ( ) 6,3429537...
E) ( ) 3,4444...
1) ( ) 0.232425...
G) ( ) 8,3333...
2) ESCREVA A FRAÇÃO GERATRIZ DAS DIZIMAS PERIODICAS SIMPLES, PARTE INTEIRA
(ZERO)
A) 0.3333...
B) 0,454545... -
C) 0,523523523...
D) 0,14141414...
O 0634634634...
F) 0,121212... -
o 0,263263263...​

Answer 1

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$\green{\rm\underline{EXPLICAC_{\!\!\!,}\tilde{A}O\ PASSO{-}A{-}PASSO\ \ \ }}$✍

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☺lá, Myllena, como tens passado nestes tempos de quarentena⁉ E os estudos à distância, como vão⁉ Espero que bem❗ Acompanhe a resolução abaixo, feita através de algumas manipulações algébricas, e após o resultado você encontrará um link com mais informações sobre Fração Geratriz que talvez te ajude com exercícios semelhantes no futuro. ✌

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$\bf\large\red{\underline{\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad}}$✍

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☔ Temos que números com dízimas periódicas (ou seja, com números decimais que se repetem em períodos, como 0,131313, que tem um período de 13 e é escrito como $0,\overline{13}$) possuem representação racional {Q}, como veremos no exercício 2, ao contrário dos irracionais {I}.

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1)$\bf\large\red{\underline{\qquad\qquad\qquad\qquad\qquad\qquad\quad}}$✍

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$\large\green{\boxed{\rm~~~\red{ A)}~\orange{\{2,354329...}~\pink{\in}~\blue{I\}}~~~}}$ ✅

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$\large\green{\boxed{\rm~~~\red{ B)}~\orange{\{3,\overline{14}}~\pink{\in}~\blue{Q\}}~~~}}$ ✅

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$\large\green{\boxed{\rm~~~\red{ C)}~\orange{\{5,\overline{324}}~\pink{\in}~\blue{Q\}}~~~}}$ ✅

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$\large\green{\boxed{\rm~~~\red{ D)}~\orange{\{6,3429537...}~\pink{\in}~\blue{I\}}~~~}}$ ✅

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$\large\green{\boxed{\rm~~~\red{ E)}~\orange{\{3,\overline{4}}~\pink{\in}~\blue{Q\}}~~~}}$ ✅

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$\large\green{\boxed{\rm~~~\red{ F)}~\orange{\{0,232425...}~\pink{\in}~\blue{I\}}~~~}}$ ✅

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$\large\green{\boxed{\rm~~~\red{ G)}~\orange{\{8,\overline{3}}~\pink{\in}~\blue{Q\}}~~~}}$ ✅

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2)Ⓐ$\bf\large\red{\underline{\qquad\qquad\qquad\qquad\qquad\qquad}}$✍

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$\large\gray{\boxed{\rm\blue{ x = 0,\overline{3} }}}$

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➡ $\sf\large\blue{ 10x - x = 9x }$

➡ $\sf\large\blue{ x = \dfrac{10x - x}{9} }$

➡ $\sf\large\blue{ x = \dfrac{3,\overline{3}  - 0,\overline{3}}{9} }$

➡ $\sf\large\blue{ x = \dfrac{3 + 0,\overline{3}  - 0,\overline{3}}{9} }$

➡ $\sf\large\blue{ x = \dfrac{3}{9} }$

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$\large\green{\boxed{\rm~~~\red{ A)}~\orange{x}~\pink{=}~\blue{ \dfrac{1}{3} }~~~}}$ ✅

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Ⓑ$\bf\large\red{\underline{\qquad\qquad\qquad\qquad\qquad\qquad\quad}}$✍

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➡ $\sf\large\blue{ 100x - x = 99x }$

➡ $\sf\large\blue{ x = \dfrac{45,\overline{45}  - 0,\overline{45}}{99} }$

➡ $\sf\large\blue{ x = \dfrac{45}{99} }$

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$\large\green{\boxed{\rm~~~\red{ B)}~\orange{x}~\pink{=}~\blue{ \dfrac{5}{11} }~~~}}$ ✅

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Ⓒ$\bf\large\red{\underline{\qquad\qquad\qquad\qquad\qquad\qquad\quad}}$✍

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➡ $\sf\large\blue{ 1000x - x = 999x }$

➡ $\sf\large\blue{ x = \dfrac{523,\overline{523}  - 0,\overline{523}}{999} }$

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$\large\green{\boxed{\rm~~~\red{ C)}~\orange{x}~\pink{=}~\blue{ \dfrac{523}{999} }~~~}}$ ✅

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Ⓓ$\bf\large\red{\underline{\qquad\qquad\qquad\qquad\qquad\qquad\quad}}$✍

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➡ $\sf\large\blue{ 100x - x = 99x }$

➡ $\sf\large\blue{ x = \dfrac{14,\overline{14}  - 0,\overline{14}}{99} }$

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$\large\green{\boxed{\rm~~~\red{ D)}~\orange{x}~\pink{=}~\blue{ \dfrac{14}{99} }~~~}}$ ✅

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Ⓔ$\bf\large\red{\underline{\qquad\qquad\qquad\qquad\qquad\qquad\quad}}$✍

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➡ $\sf\large\blue{ 1000x - x = 999x }$

➡ $\sf\large\blue{ x = \dfrac{634,\overline{634}  - 0,\overline{634}}{999} }$

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$\large\green{\boxed{\rm~~~\red{ E)}~\orange{x}~\pink{=}~\blue{ \dfrac{634}{999} }~~~}}$ ✅

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Ⓕ$\bf\large\red{\underline{\qquad\qquad\qquad\qquad\qquad\qquad\quad}}$✍

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➡ $\sf\large\blue{ 100x - x = 99x }$

➡ $\sf\large\blue{ x = \dfrac{12,\overline{12}  - 0,\overline{12}}{99} }$

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$\large\green{\boxed{\rm~~~\red{ F)}~\orange{x}~\pink{=}~\blue{ \dfrac{4}{33} }~~~}}$ ✅

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Ⓖ$\bf\large\red{\underline{\qquad\qquad\qquad\qquad\qquad\qquad\quad}}$✍

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➡ $\sf\large\blue{ 1000x - x = 999x }$

➡ $\sf\large\blue{ x = \dfrac{263,\overline{263}  - 0,\overline{263}}{999} }$

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$\large\green{\boxed{\rm~~~\red{ G)}~\orange{x}~\pink{=}~\blue{ \dfrac{263}{999} }~~~}}$ ✅

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$\bf\large\red{\underline{\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad}}$☁

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