Solve $14 \times 627 \div \sqrt{1089} = x^3 + 141$

### Concept & Formula This problem evaluates operational hierarchy (BODMAS/VBODMAS) along with finding perfect square roots and solving basic cubic equations. * Square root calculation: $\sqrt{1089} = 33$ ### Step-by-Step Solution * **Step 1:** Find the square root of $1089$: $$\sqrt{1089} = 33 \quad (\text{since } 30^2 = 900 \text{ and } 40^2 = 1600)$$ * **Step 2:** Substitute back into the equation and apply division first: $$14 \times (627 \div 33) = x^3 + 141$$ $$627 \div 33 = 19$$ * **Step 3:** Perform the multiplication: $$14 \times 19 = 266$$ * **Step 4:** Equate and solve for $x^3$: $$266 = x^3 + 141$$ $$x^3 = 266 - 141 = 125$$ * **Step 5:** Compute the cube root: $$x = \sqrt[3]{125} = 5$$ ### Exam Strategy & Shortcut **Quick Estimation:** Knowing basic squares and cubes speeds up this process tremendously. Knowing $\sqrt{1089} = 33$ simplifies $627/33$ to just under $20$ (specifically $19$). $14 \times 19$ is roughly $266$. Subtracting $141$ yields exactly $125$. Recognizing $125$ as a perfect cube ($5^3$) immediately reveals $x = 5$. ### Common Pitfall * **Wrong Order of Operations:** Executing $14 \times 627$ before dividing by $\sqrt{1089}$ creates massive numbers that are difficult to manage. Always perform division before multiplication according to BODMAS rules. ### Final Answer **Therefore, the correct answer is 5.**