Simplifying nested radical expression is discussed in the book  Power Calculation . \begin{align} &\sqrt{\sqrt[3]{9}+6\sqrt[3]{3}+9}\\ &=\sqrt{(\sqrt[3]{3})^2 + 2\times 3 \times\sqrt[3]{3} + 3^2}\\ &=\sqrt{(\sqrt[3]{3}+3)^2}\\ &=\boxed{\sqrt[3]{3}+3} \end{align}