Question

Тождественное преобразование со степенями и корнями.
$\frac{\sqrt[\frac{1}{3} ]{x} }{\sqrt[\frac{1}{3} ]{x} +\sqrt[\frac{1}{3} ]{5} }$

Answer 1

$\boxed {\ \sqrt[n]{x}=x^{\frac{1}{n}}\ }\\\\\boxed{\ (a+b)(a^2-ab+b^2)=a^3+b^3\ }\ \ ,\ \ \ \ a=x^{\frac{1}{3}}\ ,\ b=5^{\frac{1}{3}}\ ,\ \ ab=x^{\frac{1}{3}}\cdot 5^{\frac{1}{3} }=(5x)^{\frac{1}{3}}$   ,  

$\boxed {\ (x^{n})^{k}=x^{nk}\ \ \ \to \ \ \ (x^{\frac{1}{3}})^3=x\ }$

$\dfrac{\sqrt[3]{x}}{\sqrt[3]{x}+\sqrt[3]5}=\dfrac{x^{\frac{1}{3}}}{x^{\frac{1}{3}}+5^{\frac{1}{3}}}=\dfrac{x^{\frac{1}{3}}\cdot (\ x^{\frac{2}{3}}-(5x)^{\frac{1}{3}}+5^{\frac{2}{3}}\ )}{(\ x^{\frac{1}{3}}+5^{\frac{1}{5}}\ )(\ x^{\frac{2}{3}}-(5x)^{\frac{1}{3}}+5^{\frac{2}{3}}\ )}=\\\\\\=\dfrac{x^{\frac{1}{3}}\cdot (\ x^{\frac{2}{3}}-(5x)^{\frac{1}{3}}+5^{\frac{2}{3}}\ )}{x+5}=\dfrac{\sqrt[3]{x}\cdot (\sqrt[3]{x^2}-\sqrt[3]{5x}+\sqrt[3]{5^2})}{x+5}$