Solve: $(\sqrt{7} + 11)^2 = (x)^{\frac{1}{3}} + 2\sqrt{847} + 122$

Concept & Formula Expand the binomial square on the left side using the standard algebraic identity, then simplify the radicals on the right side to balance the equation and isolate $x$. $$ (a + b)^2 = a^2 + b^2 + 2ab $$ Step-by-Step Solution * **Calculation:** Expand the left-hand side (LHS): $(\sqrt{7} + 11)^2 = (\sqrt{7})^2 + (11)^2 + 2(11)(\sqrt{7})$ $= 7 + 121 + 22\sqrt{7}$ $= 128 + 22\sqrt{7}$ * Simplify the radical on the right-hand side (RHS): Notice that 847 is divisible by 7 (since $847 \div 7 = 121$). $\sqrt{847} = \sqrt{121 \times 7} = 11\sqrt{7}$ So, $2\sqrt{847} = 2(11\sqrt{7}) = 22\sqrt{7}$ * Reconstruct the equation: $128 + 22\sqrt{7} = x^{\frac{1}{3}} + 22\sqrt{7} + 122$ * Cancel out $22\sqrt{7}$ from both sides: $128 = x^{\frac{1}{3}} + 122$ * Isolate $x^{\frac{1}{3}}$: $x^{\frac{1}{3}} = 128 - 122$ $x^{\frac{1}{3}} = 6$ * Cube both sides to find $x$: $x = 6^3 = 216$ Exam Strategy & Shortcut When you see a mixed radical term like $2\sqrt{847}$ on one side and a binomial expansion with $\sqrt{7}$ on the other, anticipate that the irrational parts will cancel out. Immediately check if $847 / 7$ is a perfect square. It is (121). This instantly tells you $2\sqrt{847} = 22\sqrt{7}$, which perfectly cancels the $2ab$ term from the LHS expansion, leaving only simple integer arithmetic. Common Pitfall A major pitfall is forgetting that $x^{\frac{1}{3}}$ means the cube root of $x$, and incorrectly taking the square root or dividing by 3 at the final step. Always remember to cube the result to isolate $x$ when dealing with a one-third power. Final Answer **Therefore, the correct answer is 216.**