3. \( \left(2^{3}+3^{2}\right)-8 \cdot \sqrt{4.5-2^{2}} \)

To simplify the expression \( \left(2^{3}+3^{2}\right)-8 \cdot \sqrt{4.5-2^{2}} \), we first tackle each part step by step. First, calculate \( 2^{3} \) and \( 3^{2} \): - \( 2^{3} = 8 \) - \( 3^{2} = 9 \) Now, combine these: \[ 2^{3} + 3^{2} = 8 + 9 = 17 \] Next, simplify the expression under the square root: \[ 4.5 - 2^{2} = 4.5 - 4 = 0.5 \] Now calculate \( \sqrt{0.5} \): \[ \sqrt{0.5} = \frac{\sqrt{2}}{2} \] Multiply by 8: \[ 8 \cdot \sqrt{0.5} = 8 \cdot \frac{\sqrt{2}}{2} = 4\sqrt{2} \] Now, substitute back into the overall expression: \[ 17 - 4\sqrt{2} \] Thus, the final answer is: \[ \boxed{17 - 4\sqrt{2}} \] This expression represents the simplified result of your original input.