The problem asks to express the radical $\pm \sqrt{-77}$ using the imaginary unit $i$ and simplify the expression.

Solution:

1. The square root of a negative number involves the imaginary unit $i$, where $i = \sqrt{-1}$.
2. First, separate the negative sign inside the square root: $\sqrt{-77} = \sqrt{-1 \times 77} = \sqrt{-1} \times \sqrt{77} = i \sqrt{77}$
3. Therefore, $\pm \sqrt{-77} = \pm i \sqrt{77}$.

The simplified form is: $\boxed{\pm i \sqrt{77}}$

Would you like more details or have any questions?

Here are 5 related questions to deepen your understanding:

1. How would you express $\sqrt{-100}$ using the imaginary unit $i$?
2. What is the square root of a negative number conceptually in the complex plane?
3. How do you simplify the expression $\sqrt{-x}$ for a general negative number $x$?
4. Can you use the imaginary unit $i$ in addition and multiplication of complex numbers?
5. What are the properties of the imaginary unit $i$ in powers, such as $i^2$, $i^3$, etc.?

Tip: Always remember that $\sqrt{-1} = i$, and when working with radicals of negative numbers, factor out the $i$ first before simplifying further.