Simplify the expression to \( a+b i \) form: \[ \sqrt{9}+\sqrt{-162}+\sqrt{16}+\sqrt{-128} \]

To simplify the expression \( \sqrt{9}+\sqrt{-162}+\sqrt{16}+\sqrt{-128} \), we start by calculating each square root individually: 1. \( \sqrt{9} = 3 \). 2. Now for \( \sqrt{-162} \): since it is negative, we can factor it as \( \sqrt{-1} \cdot \sqrt{162} = i \cdot \sqrt{162} = i \cdot \sqrt{81 \cdot 2} = i \cdot 9\sqrt{2} \). 3. Next, \( \sqrt{16} = 4 \). 4. For \( \sqrt{-128} \), we apply the same method: \( \sqrt{-128} = \sqrt{-1 \cdot 128} = i \cdot \sqrt{128} = i \cdot \sqrt{64 \cdot 2} = i \cdot 8\sqrt{2} \). Now, we combine all these results back into the expression: \[ \sqrt{9}+\sqrt{-162}+\sqrt{16}+\sqrt{-128} = 3 + 9\sqrt{2} i + 4 + 8\sqrt{2} i \] Next, we combine the real parts and the imaginary parts: - Real part: \( 3 + 4 = 7 \) - Imaginary part: \( 9\sqrt{2} i + 8\sqrt{2} i = (9\sqrt{2} + 8\sqrt{2}) i = 17\sqrt{2} i \) Thus, the simplified expression in the \( a + bi \) form is: \[ \boxed{7 + 17\sqrt{2} i} \]