Question

Seja A elevado a 5 = 3, calcule se existir,

A)√a elevado a 20.
B) 3√a elevado a 15.
C)3√a elevado a 75.

Answer 1

Se $a^{5}=3,$ então

$a=\,^{5}\!\!\!\sqrt{3}\\ \\ a=3^{\frac{1}{5}}$


A) $\sqrt{a^{20}}$

$=\sqrt{(3^{\frac{1}{5})^{20}}}\\ \\ =\sqrt{3^{\frac{1}{5}\,\cdot \,20}}\\ \\ =\sqrt{3^{\frac{20}{5}}}\\ \\ =\sqrt{3^{4}}\\ \\ =\sqrt{3^{2\,\cdot\, 2}}\\ \\ =\sqrt{(3^{2})^{2}}\\ \\ =\sqrt{9^{2}}\\ \\ =9$


B) $\,^{3}\!\!\!\sqrt{a^{15}}$

$=\,^{3}\!\!\!\sqrt{(3^{\frac{1}{5}})^{15}}\\ \\ =\,^{3}\!\!\!\sqrt{3^{\frac{1}{5}\,\cdot \,15}}\\ \\ =\,^{3}\!\!\!\sqrt{3^{\frac{15}{5}}}\\ \\ =\,^{3}\!\!\!\sqrt{3^{3}}\\ \\ =3$


C) $\,^{3}\!\!\!\sqrt{a^{75}}$

$=\,^{3}\!\!\!\sqrt{(3^{\frac{1}{5}})^{75}}\\ \\ =\,^{3}\!\!\!\sqrt{3^{\frac{1}{5}\,\cdot \,75}}\\ \\ =\,^{3}\!\!\!\sqrt{3^{\frac{75}{5}}}\\ \\ =\,^{3}\!\!\!\sqrt{3^{15}}\\ \\ =\,^{3}\!\!\!\sqrt{3^{5\,\cdot\,3}}\\ \\ =\,^{3}\!\!\!\sqrt{(3^{5})^{3}}\\ \\ =3^{5}\\ \\ =243$