Questions: Perform the indicated operations. [ (-6+sqrt-16)^2 ]

Solution Steps

To find \(\sqrt{-16}\), we recognize that it can be expressed in terms of the imaginary unit \(i\): \[ \sqrt{-16} = \sqrt{16} \cdot \sqrt{-1} = 4i \]

Now, we substitute \(\sqrt{-16}\) back into the original expression: \[ (-6 + \sqrt{-16})^2 = (-6 + 4i)^2 \]

Using the formula \((a + b)^2 = a^2 + 2ab + b^2\), we expand the expression: \[ (-6 + 4i)^2 = (-6)^2 + 2(-6)(4i) + (4i)^2 \] Calculating each term: \[ (-6)^2 = 36, \quad 2(-6)(4i) = -48i, \quad (4i)^2 = 16i^2 = 16(-1) = -16 \] Combining these results gives: \[ 36 - 48i - 16 = 20 - 48i \]

Final Answer