Simplify the expression to \( a+b i \) form: \[ \sqrt{121}-\sqrt{-180}-\sqrt{36}+\sqrt{-5} \]

To simplify the expression \( \sqrt{121}-\sqrt{-180}-\sqrt{36}+\sqrt{-5} \), we will tackle each square root separately. 1. \( \sqrt{121} = 11 \) (since 121 is a perfect square). 2. For \( \sqrt{-180} \), we can write it as \( \sqrt{180} \cdot i = \sqrt{36 \cdot 5} \cdot i = 6\sqrt{5} \cdot i \). 3. \( \sqrt{36} = 6 \). 4. For \( \sqrt{-5} \), we write it as \( \sqrt{5} \cdot i \). Now, substitute these back into the original expression: \[ 11 - 6\sqrt{5} i - 6 + \sqrt{5} i \] Now, combine like terms. For the real parts: \[ 11 - 6 = 5 \] For the imaginary parts: \[ -6\sqrt{5} i + \sqrt{5} i = (-6\sqrt{5} + \sqrt{5}) i = -5\sqrt{5} i \] Putting it all together, we find: \[ 5 - 5\sqrt{5} i \] So, the expression simplifies to: \[ \boxed{5 - 5\sqrt{5} i} \]