Simplify the expression to \( a+b i \) form: \[ \sqrt{9}+\sqrt{-54}-\sqrt{36}-\sqrt{-24} \]

To simplify the expression \( \sqrt{9}+\sqrt{-54}-\sqrt{36}-\sqrt{-24} \), let's break it down step by step. 1. Calculate \( \sqrt{9} \): \[ \sqrt{9} = 3 \] 2. Calculate \( \sqrt{-54} \): \[ \sqrt{-54} = \sqrt{54} \cdot i = \sqrt{9 \cdot 6} \cdot i = 3\sqrt{6} \cdot i \] 3. Calculate \( \sqrt{36} \): \[ \sqrt{36} = 6 \] 4. Calculate \( \sqrt{-24} \): \[ \sqrt{-24} = \sqrt{24} \cdot i = \sqrt{4 \cdot 6} \cdot i = 2\sqrt{6} \cdot i \] Now, substitute these results back into the original expression: \[ 3 + 3\sqrt{6} \cdot i - 6 - 2\sqrt{6} \cdot i \] Now simplify by combining like terms: - Real parts: \( 3 - 6 = -3 \) - Imaginary parts: \( 3\sqrt{6}i - 2\sqrt{6}i = (3\sqrt{6} - 2\sqrt{6})i = \sqrt{6}i \) Thus, the expression simplifies to: \[ -3 + \sqrt{6} i \] So, in \( a + bi \) form, the simplified expression is: \[ \boxed{-3 + \sqrt{6} i} \]