First find the cube root of 729, then square that result. What final value do you obtain?

Introduction / Context:
This question connects cube roots and squares, two very basic but important operations in algebra. Problems like this are common in school level mathematics and competitive exams to test how well you understand powers and roots of numbers, especially perfect cubes and perfect squares.

Given Data / Assumptions:

• We are given the number 729.
• We must first find its cube root.
• Then we must square the cube root obtained in the first step.
• No approximation is needed because 729 is a perfect cube.

Concept / Approach:
The cube root of a number N is the value r such that r^3 = N. After finding the cube root, squaring that result means calculating r^2. For perfect powers, it helps to remember common cubes and squares: for example 3^3 = 27, 5^3 = 125, 9^3 = 729, and 9^2 = 81. Recognizing these patterns makes such questions extremely fast to solve.

Step-by-Step Solution:
Step 1: Observe that 729 can be written as 9^3, because 9 * 9 * 9 = 729. Step 2: Therefore, the cube root of 729 is 9. Step 3: Next, we must square this cube root. So we compute 9^2. Step 4: Calculate 9^2 = 9 * 9 = 81. Step 5: Hence, after taking the cube root of 729 and then squaring the result, we obtain 81.

Verification / Alternative check:
We can check in reverse. If the final answer is 81, its square root is 9. Cubing 9 gives 729, which is exactly the original number. This confirms that we have correctly identified 9 as the cube root of 729 and 81 as its square, so the overall two step process is consistent.

Why Other Options Are Wrong:
Option 9 is only the cube root of 729, not the final squared value. Option 27 could appear if someone mistakenly thought that 3 is the cube root and then computed 3^3, but 3^3 is 27, not 729. Option 144 is 12^2 and does not arise from any correct operation here. Option 36 might come from squaring 6, but 6 is not related to 729 as a cube root. Only 81 matches the described two step procedure.

Common Pitfalls:
A common mistake is to confuse square roots and cube roots, for example trying to find a number whose square is 729 instead of whose cube is 729. Another error is to stop after finding the cube root and forget to square it as the second step. Always read multi step instructions fully and complete each operation in order.

Final Answer:
Thus, the cube root of 729 is 9 and squaring this gives 81 as the final result.