Evaluate the following: \( 0.5\left[\left(0.5 \times 3^{2}+2.5\right)^{2} \div 2^{3}\right]= \) \( \square \)

Calculate the value by following steps: - step0: Calculate: \(0.5\left(\frac{\left(0.5\times 3^{2}+2.5\right)^{2}}{2^{3}}\right)\) - step1: Multiply the numbers: \(0.5\left(\frac{\left(4.5+2.5\right)^{2}}{2^{3}}\right)\) - step2: Add the numbers: \(0.5\left(\frac{7^{2}}{2^{3}}\right)\) - step3: Remove the parentheses: \(0.5\times \frac{7^{2}}{2^{3}}\) - step4: Convert the expressions: \(\frac{1}{2}\times \frac{7^{2}}{2^{3}}\) - step5: Multiply the fractions: \(\frac{1\times 7^{2}}{2\times 2^{3}}\) - step6: Multiply the terms: \(\frac{7^{2}}{2\times 2^{3}}\) - step7: Multiply the terms: \(\frac{7^{2}}{2^{4}}\) - step8: Evaluate the power: \(\frac{49}{2^{4}}\) - step9: Evaluate the power: \(\frac{49}{16}\) To evaluate the expression \[ 0.5\left[\left(0.5 \times 3^{2}+2.5\right)^{2} \div 2^{3}\right], \] let's break it down step by step: 1. **Calculate \(3^2\)**: \[ 3^2 = 9. \] 2. **Multiply by \(0.5\)**: \[ 0.5 \times 9 = 4.5. \] 3. **Add \(2.5\)**: \[ 4.5 + 2.5 = 7. \] 4. **Square the result**: \[ 7^2 = 49. \] 5. **Calculate \(2^3\)**: \[ 2^3 = 8. \] 6. **Divide \(49\) by \(8\)**: \[ 49 \div 8 = 6.125. \] 7. **Multiply by \(0.5\)**: \[ 0.5 \times 6.125 = 3.0625. \] Thus, the final result is \[ \boxed{3.0625}. \]