To find all the perfect cubes between 400 and 800, we need to determine which integers, when cubed, fall within this range.

Step-by-Step Approach:

• Start by finding the cube roots of 400 and 800.

• The cube root of 400 is approximately $\sqrt[3]{400} \approx 7.37$, and the cube root of 800 is approximately $\sqrt[3]{800} \approx 9.28$.

Thus, we are interested in the integers between 7 and 9 (inclusive).

Now, let's compute the cubes of the integers 7, 8, and 9:

• $7^3 = 343$
• $8^3 = 512$
• $9^3 = 729$

Conclusion:

The perfect cubes between 400 and 800 are:

• 512 (from $8^3$)
• 729 (from $9^3$)

Would you like further details or clarifications? Here are some additional related questions:

1. What is the largest perfect cube less than 1000?
2. How do we find the cube roots of numbers manually?
3. What are the properties of perfect cubes in number theory?
4. Can we find perfect cubes between two specific numbers using a formula?
5. How do cube roots relate to other roots like square roots?

Tip: To estimate cube roots for numbers that aren’t perfect cubes, you can use the fact that cube roots fall between whole numbers, and then refine the approximation by trying successive cubes.