Question

32 PONTOS!!! Resolva com as contas do resultado
$a)\frac{6}{ \sqrt{2} }$

$b) \frac{3}{2 \sqrt{5} }$

$c) \frac{3}{ \sqrt{2} }$

$d)\frac{5}{ \sqrt{5} }$

$e) \frac{3a}{ \sqrt[3]{a} }$

$f) \frac{5}{ \sqrt[4]{5} }$

$g) \frac{12}{ \sqrt[4]{3 {}^{3} } }$

$h) \frac{5 \sqrt{7} }{ \sqrt{7 - \sqrt{2} } }$
​

Answer 1

Resposta:

$a)\\\frac{6}{\sqrt{2}} .\frac{\sqrt{2}}{\sqrt{2}} =\frac{6}{2}.\sqrt{2} =3\sqrt{2}\\\\b)\\\frac{3}{2\sqrt{5}}=\frac{3}{2\sqrt{5}}.\frac{\sqrt{5}}{\sqrt{5}}=\frac{3\sqrt{5}}{10}\\\\c)\\\frac{3}{\sqrt{2}}=\frac{3}{\sqrt{2}}.\frac{\sqrt{2}}{\sqrt{2}}}=\frac{3}{2}\sqrt{2}\\\\d)\\\frac{5}{\sqrt{5}}=\frac{5}{\sqrt{5}}.\frac{\sqrt{5}}{\sqrt{5}}}=\sqrt{2}$

$\\e)\\\frac{3a}{\sqrt[3]{a}}=\frac{3a}{\sqrt[3]{a}}.\frac{\sqrt[3]{a^{2}}}{\sqrt[3]{a^{2}}}= \frac{3a}{\sqrt[3]{a^3}}.\sqrt[3]{a^{2}}=\frac{3a}{a^{\frac{3}{3}}}.\sqrt[3]{a^{2}}=\frac{3a}{a}.\sqrt[3]{a^{2}}=3.\sqrt[3]{a^{2}}\\\\f)\\\frac{5}{\sqrt[4]{5}}=\frac{5}{\sqrt[4]{5}}.\frac{\sqrt[4]{5^{3}}}{\sqrt[4]{5^{3}}} =\sqrt[4]{5^{3}}=\sqrt[4]{125} \\\\g)\\\frac{12}{\sqrt[4]{3^{3}}}=\frac{12}{\sqrt[4]{3^{3}}}.\frac{\sqrt[4]{3}}{\sqrt[4]{3}}=\frac{12}{3}\sqrt[4]{3}=4\sqrt[4]{3}$

$h)\\\frac{5\sqrt{7}}{\sqrt{7-\sqrt{2}}}=\frac{5\sqrt{7}}{\sqrt{7-\sqrt{2}}}.\frac{\sqrt{7+\sqrt{2}}}{\sqrt{7+\sqrt{2}}}=\frac{5\sqrt{7}\sqrt{7+\sqrt{2}}}{{\sqrt{7^{2}-(\sqrt{2})^{2}}}}=\frac{5\sqrt{7}\sqrt{7+\sqrt{2}}}{{\sqrt{49-2}}} =\frac{5\sqrt{7}\sqrt{7+\sqrt{2}}}{{\sqrt{47}}}.\frac{\sqrt{47}}{\sqrt{47}}=\\\\=\frac{5\sqrt{7.47}\sqrt{7+\sqrt{2}}}{{47}}=\frac{5\sqrt{329}\sqrt{7+\sqrt{2}}}{{47}}$

Propriedades:

$(a^{m})^{n}=a^{m.n}\\\\\sqrt[n]{x^m} =x^{\frac{m}{n} }\\\\a^{m}a^{n}=a^{m+n}\\\\\frac{a^{m}}{a^{n}}=a^{m-n} \\\\a^{0}=1\\\\a^{1}=a$