To solve for $w$ in the equation:

$w^2 = -81$

we start by isolating $w$. To do this, take the square root of both sides of the equation:

$w = \pm \sqrt{-81}$

Since the square root of a negative number involves imaginary numbers, we rewrite $\sqrt{-81}$ as:

$w = \pm \sqrt{81} \cdot \sqrt{-1} = \pm 9i$

where $i$ is the imaginary unit, defined as $i = \sqrt{-1}$.

Final Answer:

$w = 9i \quad \text{or} \quad w = -9i$

Would you like further details, or do you have any questions?

Here are some related questions that expand on this concept:

1. What are imaginary numbers and how are they used in complex solutions?
2. How can we verify that $w = \pm 9i$ satisfies the original equation $w^2 = -81$?
3. How do square roots work when involving negative numbers?
4. What is the difference between real and complex solutions?
5. How would we solve an equation like $w^2 = 81$ (without the negative)?

Tip: When dealing with square roots of negative numbers, remember to use the imaginary unit $i$ to express the solution properly.