Question

indece 8 que seja equivalente a ¹²√5²¹

Answer 1

Transformando a raiz para uma potência com expoente fracionário:

$\,^{12}\!\!\!\sqrt{5^{21}}=5^{21/12}=5^{x/8}$

Encontrar o valor de $x:$

$\dfrac{21}{12}=\dfrac{x}{8}\\\\\\ \dfrac{\diagup\!\!\!\! 3\cdot 7}{\diagup\!\!\!\! 3\cdot 4}=\dfrac{x}{8}\\\\\\ \dfrac{7}{4}=\dfrac{x}{8}\\\\\\ 4x=7\cdot 8\\\\ 4x=56\\\\ x=\dfrac{56}{4}\\\\\\ \boxed{\begin{array}{c}x=14 \end{array}}$

Portanto,

$5^{21/12}=5^{14/8}\\\\\\ \therefore~~\boxed{\begin{array}{c}\,^{12}\!\!\!\sqrt{5^{21}}=\,^{8}\!\!\!\sqrt{5^{14}} \end{array}}$