Question

1) calcule os radicais a) raiz quadrada de 27 : raiz quadrada de 3 b) raiz cubica de 4 : raiz quadrada de 2 c) raiz quadrada de 2 . raiz quadrada de 10 . raiz cubica de 5. ( ":" = a dividir)​

Answer 1

a)   $\frac{\sqrt{27}}{\sqrt{3}}=$

$=\frac{\sqrt{3^{3}}\times\sqrt{3}}{\sqrt{3}\times\sqrt{3}}=$

$=\frac{\sqrt{3^{2}\times3\times3}}{(\sqrt{3})^{2}}=$

$=\frac{\sqrt{3^{2}\times3^{2}}}{3}=$

$=\frac{3\times3}{3}=$

$=\frac{9}{3}=$

$=3$

b)   $\frac{\sqrt[3]{4}}{\sqrt{2}}=$

$=\frac{\sqrt[3]{4}\times\sqrt{2}}{\sqrt{2}\times\sqrt{2}}=$

$=\frac{\sqrt[3]{2^{2}}\times\sqrt{2}}{(\sqrt{2})^{2}}=$

$=\frac{2^{\frac{2}{3}}\times2^{\frac{1}{2}}}{2}=$

$=\frac{2^{\frac{2}{3}+\frac{1}{2}}}{2}=$

$=\frac{2^{\frac{2\times2}{3\times2}+\frac{3}{2\times3}}}{2}=$

$=\frac{2^{\frac{4}{6}+\frac{3}{6}}}{2}=$

$=2^{\frac{7}{6}-1}=$

$=2^{\frac{7}{6}-\frac{6}{6}}=$

$=2^{\frac{1}{6}}=$

$=\sqrt[6]{2}$

c)   $\sqrt{2}\times\sqrt{10}\times\sqrt[3]{5}=$

$=\sqrt{2}\times\sqrt{2\times5}\times\sqrt[3]{5}=$

$=\sqrt{2}\times\sqrt{2}\times\sqrt{5}\times\sqrt[3]{5}=$

$=(\sqrt{2})^{2}\times5^{\frac{1}{2}}\times5^{\frac{1}{3}}=$

$=2\times5^{\frac{1}{2}+\frac{1}{3}}=$

$=2\times5^{\frac{3}{2\times3}+\frac{2}{3\times2}}=$

$=2\times5^{\frac{3}{6}+\frac{2}{6}}=$

$=2\times5^{\frac{5}{6}}=$

$=2\sqrt[6]{5^{5}}=$

$=2\sqrt[6]{3125}}$

Answer 2

Explicação passo-a-passo:

Divisão de Radicais :

$\mathtt{ A)~ \dfrac{ \sqrt{27} }{ \sqrt{3} } }$

$\mathtt{ \dfrac{ \sqrt{27}}{\sqrt{3}}~=~ \sqrt{ \dfrac{27}{3} } }$

$\mathtt{ \dfrac{ \sqrt{27}}{ \sqrt{ 3} }~=~\sqrt{9} }$

$\mathtt{ \dfrac{ \sqrt{27} }{ \sqrt{3} }~=~\red{3} }$

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B)

$\mathtt{ B)~\dfrac{ \sqrt[3]{4} }{ \sqrt{2} } }$

$\mathtt{ \dfrac{ \sqrt[3]{4} }{ \sqrt{2} }~=~\dfrac{ 2^{\frac{2}{3} } }{ 2^{\frac{1}{2} } } }$

$\mathtt{ \dfrac{ \sqrt[3]{4} }{ \sqrt{2} } ~=~2^{\frac{2}{3} - \frac{1}{2} } ~=~2^{\frac{4-3}{6} } }$

$\mathtt{ \dfrac{ \sqrt[3]{4} }{ \sqrt{2} }~=~2^{\frac{1}{6}} ~=~\red{ \sqrt[6]{2} } }$

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C)

$\mathtt{ \sqrt{2} \cdot \sqrt{10} \cdot \sqrt[3]{5} }$

$\mathtt{=~\sqrt{2 \cdot 10} \cdot \sqrt[3]{5} }$

$\mathtt{ =~\sqrt{20} \cdot \sqrt[3]{5} }$

$\mathtt{=~2\sqrt{5} \cdot \sqrt[3]{5} }$

$\mathtt{~=~2\cdot 5^{\frac{1}{2}} \cdot 5^{\frac{1}{3}} }$

$\mathtt{=~2 \cdot 5^{\frac{5}{6}} ~=~2 \cdot \sqrt[6]{5^5} }$

Espero ter ajudado bastante!)