Final answer: \(9^3 = \boxed{729}\), \(\left(\frac{1}{4}\right)^3 = \boxed{\frac{1}{64}}\), Sum of smaller areas \(\boxed{\text{not equal}}\) to the large area.
We begin by evaluating the given powers. The expression \(9^3\) represents 9 multiplied by itself three times, which is \(9 \times 9 \times 9 = 81 \times 9 = 729\). For the second expression, \(\left(\frac{1}{4}\right)^3\), we multiply the fraction \(\frac{1}{4}\) by itself three times, giving \(\frac{1}{4} \times \frac{1}{4} \times \frac{1}{4} = \frac{1}{64}\), which as a decimal is \(0.015625\).
Next, we address the question about square areas. We are given three square side lengths: 8 feet, 5 feet, and 3 feet. The area of a square is calculated as \(A = s^2\), where \(s\) is the side length.
For the two smaller squares with sides 5 and 3 feet:
The area of the first square is \(A_1 = 5^2 = 25\) square feet.
The area of the second square is \(A_2 = 3^2 = 9\) square feet.
The sum of these two areas is \(A_1 + A_2 = 25 + 9 = 34\) square feet.
The area of the larger square with side length 8 feet is \(A_{large} = 8^2 = 64\) square feet.
Comparing the two results, we see that \(34
eq 64\). Therefore, the sum of the areas of the two smaller squares is not equal to the area of the larger square.
Understanding powers and areas is foundational for geometry and physics. For example, when you calculate the floor area of a room to buy carpet or determine how much paint you need, you are using the area formula \(A = s^2\). Similarly, volume calculations, such as determining how many small cubes fit into a larger box, rely on powers like \(s^3\). Being able to compute these values accurately helps in planning home renovations, construction projects, and even simple organizational tasks.