The equation given in the image is:

$a^3 = -729$

To find $a$, we need to compute the cube root of $-729$:

$a = \sqrt[3]{-729}$

Since the cube root of $-729$ is $-9$, we have:

$a = -9$

Would you like a detailed explanation of how to compute cube roots or have any other questions?

Here are 5 related questions you might want to explore:

1. What is the cube root of a positive number, like $729$?
2. What are cube roots and how are they different from square roots?
3. Can cube roots be calculated for negative numbers, and why?
4. What are some methods to manually compute cube roots of larger numbers?
5. How do we solve equations involving higher powers, like $a^4 = 81$?

Tip: For any number $n$, the cube root can be expressed as $\sqrt[3]{n}$, which can be positive or negative depending on the sign of $n$.