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