We are given the equation:

$43 = a^3$

To solve for $a$, take the cube root of both sides:

$a = \sqrt[3]{43}$

The cube root of 43 does not simplify further, so the solution is:

$a = \sqrt[3]{43}$

This is the simplest radical form.

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

Here are some related questions:

1. What is the cube root of 64, and how does it compare to $\sqrt[3]{43}$?
2. How can cube roots be approximated using a calculator?
3. What are the cube roots of other numbers between 1 and 50?
4. What is the difference between square roots and cube roots?
5. How do cube roots relate to the volume of cubes?

Tip: Cube roots are the inverse operation of cubing a number, and unlike square roots, they work for both positive and negative numbers.