Simplify the expression to \( a+b i \) form: \[ \sqrt{64}+\sqrt{-108}-\sqrt{100}+\sqrt{-27} \]

To simplify the expression \( \sqrt{64}+\sqrt{-108}-\sqrt{100}+\sqrt{-27} \), we begin by calculating each square root. 1. \( \sqrt{64} = 8 \) 2. \( \sqrt{-108} = \sqrt{108} \cdot i = \sqrt{36 \cdot 3} \cdot i = 6\sqrt{3} \cdot i \) 3. \( -\sqrt{100} = -10 \) 4. \( \sqrt{-27} = \sqrt{27} \cdot i = \sqrt{9 \cdot 3} \cdot i = 3\sqrt{3} \cdot i \) Now, we can put these values into the original expression: \[ 8 + 6\sqrt{3} \cdot i - 10 + 3\sqrt{3} \cdot i \] Combining the real parts: \[ 8 - 10 = -2 \] And the imaginary parts: \[ 6\sqrt{3} \cdot i + 3\sqrt{3} \cdot i = (6\sqrt{3} + 3\sqrt{3}) \cdot i = 9\sqrt{3} \cdot i \] Putting it all together, we have: \[ -2 + 9\sqrt{3} \cdot i \] Thus, the expression in the \( a + bi \) form is: \[ \boxed{-2 + 9\sqrt{3} i} \]