Question

1) Introduza os fatores externos de cada um dos radicais no radicando:
 $d)\: \frac{2}{5} \sqrt{ \frac{2a^2}{c} } \\ \\ e)\: 3a^2b \sqrt[3]{ab^2} \\ \\ f)\: \frac{5a}{2b} \sqrt{ \frac{b}{a} }$

Answer 1

Veja:

$a\sqrt[n]{b}=\sqrt[n]{a^{n}}*\sqrt[n]{b}\\\\\boxed{\boxed{a\sqrt[n]{b}=\sqrt[n]{a^{n}*b}}}$

Vamos usar essa "fórmula":

d)

$\frac{2}{5}\sqrt{\frac{2a^{2}}{c}}=\frac{2}{5}\sqrt[2]{\frac{2a^{2}}{c}}\\\frac{2}{5}\sqrt{\frac{2a^{2}}{c}}=\sqrt[2]{(\frac{2}{5})^{2}}*\sqrt[2]{\frac{2a^{2}}{c}}\\ \frac{2}{5}\sqrt{\frac{2a^{2}}{c}}=\sqrt[2]{(\frac{2}{5})^{2}*\frac{2a^{2}}{c}}\\\\\frac{2}{5}\sqrt{\frac{2a^{2}}{c}}=\sqrt{\frac{2^{2}}{5^{2}}*\frac{2a^{2}}{c}}\\\frac{2}{5}\sqrt{\frac{2a^{2}}{c}}=\sqrt[2]{\frac{4}{25}*\frac{2a^{2}}{c}}\\\frac{2}{5}\sqrt{\frac{2a^{2}}{c}}=\sqrt[2]{\frac{4*2a^{2}}{25*c}}$
$\frac{2}{5}\sqrt{\frac{2a^{2}}{c}}=\sqrt{\frac{8a^{2}}{25c}}$

e)

$3a^{2}b\sqrt[3]{ab^{2}}=\sqrt[3]{(3a^{2}b)^{3}}*\sqrt[3]{ab^{2}}\\3a^{2}b\sqrt[3]{ab^{2}}=\sqrt[3]{3^{3}a^{6}b^{3}}*\sqrt[3]{ab^{2}}\\3a^{2}b\sqrt[3]{ab^{2}}=\sqrt[3]{27a^{6}b^{3}}*\sqrt[3]{ab^{2}}\\3a^{2}b\sqrt[3]{ab^{2}}=\sqrt[3]{27a^{6}b^{3}*ab^{2}}\\3a^{2}b\sqrt[3]{ab^{2}}=\sqrt[3]{27a^{7}b^{5}}$

f)

$\frac{5a}{2b}\sqrt{\frac{b}{a}}=\sqrt{(\frac{5a}{2b})^{2}}*\sqrt{\frac{b}{a}}\\\frac{5a}{2b}\sqrt{\frac{b}{a}}=\sqrt{\frac{5^{2}a^{2}}{2^{2}b^{2}}}*\sqrt{\frac{b}{a}}\\\frac{5a}{2b}\sqrt{\frac{b}{a}}=\sqrt{\frac{25a^{2}}{4b^{2}}}*\sqrt{\frac{b}{a}}\\\frac{5a}{2b}\sqrt{\frac{b}{a}}=\sqrt{\frac{25a^{2}*b}{4b^{2}*a}}\\\frac{5a}{2b}\sqrt{\frac{b}{a}}=\sqrt{\frac{25a^{1}*b^{0}}{4b^{1}*a^{0}}}\\\frac{5a}{2b}\sqrt{\frac{b}{a}}=\sqrt{\frac{25a}{4b}}$