Obliczamy wartość wyrażenia.

$\frac{\sqrt[4]{7\sqrt[3]{54}+15\sqrt[3]{128}}}{\sqrt[3]{4\cdot \sqrt[4]{32}}+\sqrt[3]{9\cdot \sqrt[4]{162}}}=\frac{\sqrt[4]{7\sqrt[3]{2\cdot 3^3}+15\sqrt[3]{2^7}}}{\sqrt[3]{4\cdot \sqrt[4]{2^5}}+\sqrt[3]{9\cdot \sqrt[4]{2\cdot 3^4}}}=$  

$=\frac{\sqrt[4]{7\cdot \sqrt[3]{2}\cdot \sqrt[3]{3^3}+15\cdot \sqrt[3]{2^6\cdot 2}}}{\sqrt[3]{4\cdot \sqrt[4]{2^4\cdot 2}}+\sqrt[3]{9\cdot \sqrt[4]{2}\cdot \sqrt[4]{3^4}}}=\frac{\sqrt[4]{7\cdot \sqrt[3]{2}\cdot 3+15\cdot \sqrt[3]{{\left(2^2\right)}^3\cdot 2}}}{\sqrt[3]{4\cdot \sqrt[4]{2^4}\cdot \sqrt[4]{2}}+\sqrt[3]{9\cdot \sqrt[4]{2}\cdot 3}}=$  

$=\frac{\sqrt[4]{21\sqrt[3]{2}+15\cdot \sqrt[3]{{\left(2^2\right)}^3}\cdot \sqrt[3]{2}}}{\sqrt[3]{4\cdot 2\cdot \sqrt[4]{2}}+\sqrt[3]{27\cdot \sqrt[4]{2}}}=\frac{\sqrt[4]{21\sqrt[3]{2}+15\cdot 2^2\cdot \sqrt[3]{2}}}{\sqrt[3]{8\sqrt[4]{2}}+\sqrt[3]{27\sqrt[4]{2}}}=$  

$=\frac{\sqrt[4]{21\sqrt[3]{2}+15\cdot 4\cdot \sqrt[3]{2}}}{\sqrt[3]{8}\cdot \sqrt[3]{\sqrt[4]{2}}+\sqrt[3]{27}\cdot \sqrt[3]{\sqrt[4]{2}}}=\frac{\sqrt[4]{21\sqrt[3]{2}+60\sqrt[3]{2}}}{\sqrt[3]{2^3}\cdot \sqrt[3]{\sqrt[4]{2}}+\sqrt[3]{3^3}\cdot \sqrt[3]{\sqrt[4]{2}}}=$  

$=\frac{\sqrt[4]{81\sqrt[3]{2}}}{2\sqrt[3]{\sqrt[4]{2}}+3\sqrt[3]{\sqrt[4]{2}}}=\frac{\sqrt[4]{81}\cdot \sqrt[4]{\sqrt[3]{2}}}{5\sqrt[3]{\sqrt[4]{2}}}=\frac{\sqrt[4]{3^4}\cdot \sqrt[4]{\sqrt[3]{2}}}{5\sqrt[3]{\sqrt[4]{2}}}=\frac{3\sqrt[4]{\sqrt[3]{2}}}{5\sqrt[3]{\sqrt[4]{2}}}=\frac{3}{5}$