Question

Poderiam me ajudar ?

1- Resolva:

$\sqrt[6]{2} ^{34} b) \sqrt[5]{7} ^{18}$

Answer 1

$\mathbf{a)~~}\,^{6}\!\!\!\sqrt{2^{34}}\\ \\ =\,^{6}\!\!\!\sqrt{2^{30+4}}\\ \\ =\,^{6}\!\!\!\sqrt{2^{30}\cdot 2^{4}}\\ \\ =\,^{6}\!\!\!\sqrt{2^{5\,\cdot\,6}\cdot 2^{4}}\\ \\ =\,^{6}\!\!\!\!\sqrt{(2^{5})^{6}\cdot 2^{4}}\\ \\ =\,^{6}\!\!\!\!\sqrt{(2^{5})^{6}}\cdot \,^{6}\!\!\!\sqrt{2^{4}}\\ \\ =2^{5}\cdot \,^{6}\!\!\!\sqrt{2^{4}}\\ \\ =32\,^{6}\!\!\!\sqrt{16}$


$\mathbf{b)~~}\,^{5}\!\!\!\sqrt{7^{18}}\\ \\ =\,^{5}\!\!\!\sqrt{7^{15+3}}\\ \\ =\,^{5}\!\!\!\sqrt{7^{15}\cdot 7^{3}}\\ \\ =\,^{5}\!\!\!\sqrt{7^{3\,\cdot\,5}\cdot 7^{3}}\\ \\ =\,^{5}\!\!\!\sqrt{(7^{3})^{5}\cdot 7^{3}}\\ \\ =\,^{5}\!\!\!\!\sqrt{(7^{3})^{5}}\cdot \,^{5}\!\!\!\sqrt{7^{3}}\\ \\ =7^{3}\cdot \,^{5}\!\!\!\sqrt{7^{3}}\\ \\ =343\,^{5}\!\!\!\sqrt{343}$