To find the value of $t$, we can use the formula for the volume of a cube, which is:

$V = t^3$

Given that the volume of the cube is 512 cubic centimeters, we can set up the equation:

$t^3 = 512$

Now, solve for $t$ by taking the cube root of both sides:

$t = \sqrt[3]{512}$

Since $512 = 8^3$, we have:

$t = 8 \text{ centimeters}$

Therefore, the value of $t$ is $8$ centimeters.

Would you like further details or have any questions?

Here are 5 related questions to consider:

1. What is the formula for the surface area of a cube with side length $t$?
2. If the side length $t$ was doubled, what would the new volume be?
3. How would the volume change if $t$ was halved?
4. What is the cube root of 729? How does it relate to finding the side length of a cube with volume 729?
5. Can you express the volume formula for a rectangular prism instead of a cube?

Tip: To solve cube volume problems, always remember to take the cube root to find the side length.