We first evaluate the expression \(9^3\), which represents 9 cubed. This is calculated as \(9 \times 9 \times 9\). Multiplying the first two terms gives \(9 \times 9 = 81\). Then, multiplying this result by 9 gives \(81 \times 9 = 729\). Thus, \(9^3 = 729\).
Next, we calculate \(\left(\frac{1}{4}\right)^3\). This expression is equivalent to \(\frac{1^3}{4^3}\). Since \(1^3 = 1\) and \(4^3 = 4 \times 4 \times 4 = 16 \times 4 = 64\), the expression becomes \(\frac{1}{64}\). In decimal form, this is \(0.015625\).
Finally, we consider the claim regarding the areas of squares with side lengths 8, 5, and 3 feet. The area of a square with side length \(s\) is \(A = s^2\). For the smaller squares, the areas are \(5^2 = 25\) and \(3^2 = 9\). Their sum is \(25 + 9 = 34\) square feet. For the large square with side length 8, the area is \(8^2 = 64\) square feet. Comparing the two values, \(34 eq 64\), so the sum of the areas of the smaller squares is not equal to the area of the large square.
Understanding how to calculate areas is essential in construction and design projects. For instance, if you are planning to tile a floor with different-sized square tiles, knowing the area covered by each tile size helps you determine the total number of tiles needed. Similarly, knowing how areas behave (such as the sum of areas of squares) helps in spatial planning and material estimation, ensuring your calculations reflect the actual space available.