ABC26GN1482 · Square Roots and Cube Roots

Let the number be $a$. $$\begin{aligned} & (\sqrt{7}+11)^{2} \\ & =a^{\frac{1}{3}}+2 \sqrt{847}+122 \\ & \Rightarrow 7+121+22 \sqrt{7} \\ & =a^{\frac{1}{3}}+22 \sqrt{7}+122 \\ & \Rightarrow 128-122=a^{\frac{1}{3}} \\ & \Rightarrow a^{\frac{1}{3}}=6 \\ & \Rightarrow a=(6)^{3}=216 \end{aligned}$$ Hence, the number is 216.