GEOMETRY A large square has an area of \( \frac{500}{3} \mathrm{~m}^{2} \) and a small square has an area of \( \frac{256}{3} \mathrm{~m}^{2} \). The side length of the large square is \( \sqrt{\frac{500}{3}} \) meters. The side length of the small square is \( \sqrt{\frac{256}{3}} \) meters. Find the difference of the side length of the large square and the side length of the small square. \( \qquad \) meters \( \square \)

To find the difference between the side lengths of the large square and the small square, we first calculate the side length for both squares. 1. For the large square, the side length is: \[ s_{\text{large}} = \sqrt{\frac{500}{3}} \, \text{meters} \] 2. For the small square, the side length is: \[ s_{\text{small}} = \sqrt{\frac{256}{3}} \, \text{meters} \] Now, we can calculate the difference in their side lengths as follows: \[ \text{Difference} = s_{\text{large}} - s_{\text{small}} = \sqrt{\frac{500}{3}} - \sqrt{\frac{256}{3}} \] Next, let's find a common value for the difference: 1. Combine the radicals: \[ = \frac{1}{\sqrt{3}} \left( \sqrt{500} - \sqrt{256} \right) \] 2. Calculate \(\sqrt{500}\) and \(\sqrt{256}\): - \(\sqrt{500} = 10\sqrt{5}\) - \(\sqrt{256} = 16\) So now we substitute back: \[ \text{Difference} = \frac{1}{\sqrt{3}} \left( 10\sqrt{5} - 16 \right) \] This gives us the final answer for the difference in side lengths of the squares, which simplifies to: \[ \text{Difference} \approx 1.837 \text{ meters} (after calculating exact values if desired). \] Thus, the difference in meters is: \[ \text{Difference} \approx 1.837 \text{ meters} \]