Question

Aplicando as propriedas , simplifique os radicais:
e) ³√54

C) ⁶√3²⁰

d) √20

f) ³√-2160

#Obs: valendo 23 Pontos :D

Answer 1

a) $\sqrt[3]{54}\\ \\ =\sqrt[3]{2\cdot 3^{3}}\\ \\ =\sqrt[3]{2}\cdot \sqrt[3]{3^{3}}\\ \\ =\sqrt[3]{2}\cdot 3\\ \\ =3\sqrt[3]{2}$ b) $\sqrt[6]{3^{20}}\\ \\ =\sqrt[6]{3^{18+2}}\\ \\ =\sqrt[6]{3^{18}\cdot3^{2}}\\ \\ =\sqrt[6]{3^{18}}\cdot\sqrt[6]{3^{2}}\\ \\ =\sqrt[6]{3^{3\,\cdot\,6}}\cdot\sqrt[6]{9}\\ \\ =\sqrt[6]{\left(3^{3} \right )^{6}}\cdot\sqrt[6]{9}\\ \\ =3^{3} \cdot\sqrt[6]{9}\\ \\ =27\sqrt[6]{9}$ c) $\sqrt{20}\\ \\ =\sqrt{2^{2}\cdot 5}\\ \\ =\sqrt{2^{2}}\cdot \sqrt{5}\\ \\ =2\sqrt{5}$ d) $\sqrt[3]{-2\,160}$ Vamos decompor o número $2\,160$ em seus fatores primos: $\begin{array}{r|l} 2\,160&2\\ 1\,080&2\\ 540&2\\ 270&2\\ 135&3\\ 45&3\\ 15&3\\ 5&5\\ 1 \end{array}\\ \\ \\ -2\,160=-2^{4}\cdot 3^{3}\cdot 5$ $\sqrt[3]{-2\,160}\\ \\ =\sqrt[3]{-2^{4}\cdot 3^{3}\cdot 5}\\ \\ =\sqrt[3]{-2^{3+1}\cdot 3^{3}\cdot 5}\\ \\ =\sqrt[3]{-2^{3}\cdot 2\cdot 3^{3}\cdot 5}\\ \\ =\sqrt[3]{-2^{3}\cdot 3^{3}\cdot 2\cdot 5}\\ \\ =\sqrt[3]{\left(-1 \right )}\cdot \sqrt[3]{2^{3}}\cdot \sqrt{3^{3}}\cdot \sqrt[3]{2\cdot 5}\\ \\ =-2\cdot 3\cdot \sqrt[3]{10}\\ \\ =-6\sqrt[3]{10}$ Então $\sqrt[3]{-2\,160}\\ \\ =\sqrt[3]{-2^{4}\cdot 3^{3}\cdot 5}\\ \\ =\sqrt[3]{-2^{3+1}\cdot 3^{3}\cdot 5}\\ \\ =\sqrt[3]{-2^{3}\cdot 2\cdot 3^{3}\cdot 5}\\ \\ =\sqrt[3]{-2^{3}\cdot 3^{3}\cdot 2\cdot 5}\\ \\ =\sqrt[3]{\left(-1 \right )}\cdot \sqrt[3]{2^{3}}\cdot \sqrt{3^{3}}\cdot \sqrt[3]{2\cdot 5}\\ \\ =-2\cdot 3\cdot \sqrt[3]{10}\\ \\ =-6\sqrt[3]{10}$