Question

Sabendo que
$x=\dfrac{3+\sqrt{6}}{5\sqrt{3}-2\sqrt{12}-\sqrt{32}+\sqrt{50}}$

Qual o valor da expressão abaixo?
$\dfrac{x^2}{\sqrt[6]{x^7}}$


Opções:
$\begin{array}{l}a)\quad\sqrt[5]{3^4}\\{}\\{}b)\quad\sqrt[7]{3^6}\\{}\\{}c)\quad\sqrt[10]{3^7}\\{}\\{}d)\quad\sqrt[12]{3^5}\end{array}$

Answer 1

Realizando os cálculos podemos concluir que o valor da expressão é:

$\boxed{\boxed{\large\displaystyle\text{ \mathsf{\sqrt[12]{3^{5}}} }}}$

Conforme a alternativa D.

Resolução do exercício

Primeiramente, devemos simplificar a expressão.

$\large\displaystyle\text{ \mathsf{x=\dfrac{3+\sqrt{6} }{5\sqrt{3}-2\sqrt{12}-\sqrt{32}+\sqrt{50}}} }\\\\\\\\\large\displaystyle\text{ \mathsf{x=\dfrac{3+\sqrt{6} }{5\sqrt{3}-2\sqrt{2^{2}.3}-\sqrt{4^{2}.2}+\sqrt{5^{2}.2}}} }\\\\\\\\\large\displaystyle\text{ \mathsf{x=\dfrac{3+\sqrt{6} }{5\sqrt{3}-4\sqrt{3}-4\sqrt{2}+5\sqrt{2}}} }\\\\\\\\\large\displaystyle\text{ \mathsf{x=\dfrac{3+\sqrt{6} }{\sqrt{3}+\sqrt{2}}} }$

Racionalizando o denominador:

$\large\displaystyle\text{ \mathsf{x=\dfrac{3+\sqrt{6} }{\sqrt{3}+\sqrt{2}}} }\\\\\\\\\large\displaystyle\text{ \mathsf{x=\dfrac{(3+\sqrt{6})~.~(\sqrt{3}-\sqrt{2})}{(\sqrt{3}+\sqrt{2})~.~(\sqrt{3}-\sqrt{2})}} }\\\\\\\\\large\displaystyle\text{ \mathsf{x=\dfrac{3\sqrt{3}-3\sqrt{2}+\sqrt{18}-\sqrt{12}}{3-\sqrt{6}+\sqrt{6}-2}} }\\\\\\\\\large\displaystyle\text{ \mathsf{x=\dfrac{3\sqrt{3}-3\sqrt{2}+\sqrt{3^{2}.2}-\sqrt{2^{2}.3}}{1}} }$

$\large\displaystyle\text{ \mathsf{x=3\sqrt{3}-3\sqrt{2}+3\sqrt{2}-2\sqrt{3}} }\\\boxed{\large\displaystyle\text{ \mathsf{x=\sqrt{3}} }}$

Sendo x =√3, podemos substituir na expressão:

$\large\displaystyle\text{ \mathsf{\dfrac{x^{2}}{\sqrt[6]{x^{7}}}} }\\\\\\\large\displaystyle\text{ \mathsf{\dfrac{\sqrt{3}^{2}}{\sqrt[6]{\sqrt{3}^{7}}}} }\\\\\\\large\displaystyle\text{ \mathsf{\dfrac{\sqrt{3}^{2}}{\sqrt[6]{\sqrt{3}.\sqrt{3}^{6}}}} }\\\\\\\\\large\displaystyle\text{ \mathsf{\dfrac{\sqrt{3}.\sqrt{3}}{\sqrt{3}.\sqrt[6]{\sqrt{3}}}} }\\\\\\\\\large\displaystyle\text{ \mathsf{\dfrac{\sqrt{3}}{\sqrt[6]{\sqrt{3}}}} }$

$\large\displaystyle\text{ \mathsf{\dfrac{(\sqrt[6]{\sqrt{3}})^{6}}{\sqrt[6]{\sqrt{3}}}} }\\\\\\\\\large\displaystyle\text{ \mathsf{\dfrac{\sqrt[6]{\sqrt{3}}~.~(\sqrt[6]{\sqrt{3}})^{5}}{\sqrt[6]{\sqrt{3}}}} }\\\\\\\\\large\displaystyle\text{ \mathsf{(\sqrt[6]{\sqrt{3}})^{5}} }\\\\\\\boxed{\large\displaystyle\text{ \mathsf{\sqrt[12]{3^{5}}} }}$

⭐ Espero ter ajudado! ⭐

Veja mais exercícios semelhantes em:

https://brainly.com.br/tarefa/53963461

Answer 2

O valor da expressão é:
$\Large \text { \dfrac{x^2}{\sqrt[6]{x^7}} = \sqrt [12]{3^{^5}} }$  ⟹ Alternativa D.

• Inicialmente simplifique a expressão numérica que representa o valor de x.

$\large \text { x=\dfrac{3+\sqrt{6}}{5\sqrt{3}-2\sqrt{12}-\sqrt{32}+\sqrt{50}} }$

• Para transformar e simplificar e algumas radiciações, observe que:

$\large 3 = \sqrt 3 \cdot \sqrt 3$

$\large \sqrt 6 = \sqrt 2 \cdot \sqrt 3$

$\large 2 \sqrt{1 2} = 2 \cdot \sqrt {4 \cdot 3} = 4 \cdot \sqrt {3}$

$\large \sqrt{32} = \cdot \sqrt {16 \cdot 2} = 4 \cdot \sqrt {2}$

$\large \sqrt{50} = \cdot \sqrt {25 \cdot 2} = 5 \cdot \sqrt {2}$

• Substitua as simplificações na equação de x.

$\large \text { x=\dfrac{\sqrt 3 \cdot \sqrt 3+\sqrt2 \cdot \sqrt 3}{5\sqrt{3}-4\sqrt{3}-4 \sqrt{2}+5 \sqrt{2}} }$

• Fatore o numerador e reduza os termos semelhantes no denominador.

$\large \text { x=\dfrac{\sqrt 3 \cdot (\sqrt 2+\sqrt 3)}{\sqrt{3}+ \sqrt{2}} \quad \Longrightarrow \quad \sf Simplifique.}$

$\large x=\sqrt 3$

• Substitua o valor de x na expressão desejada e aplique as propriedades da radiciação e potenciação.

$\large\text{ \dfrac{x^2}{\sqrt[6]{x^7}} = \dfrac{{(\sqrt 3)}^2}{\sqrt[6]{(\sqrt 3)^7}} =\dfrac{3}{\left(3^{^{\frac{1}{2}}} \right)^{\frac{7}{6}}}=\dfrac{3}{3^{^{\frac{7}{12}}}} = 3^{^{\left( 1-\frac {7}{12} \right) }} = 3^{^{\frac {5}{12}}} }$

$\Large \text { \dfrac{x^2}{\sqrt[6]{x^7}} = \sqrt [12]{3^{^5}} }$  Alternativa D.

Aprenda mais:

• brainly.com.br/tarefa/39398372
• brainly.com.br/tarefa/41200582
• brainly.com.br/tarefa/41279712
• brainly.com.br/tarefa/38206741