Question

How many perfect cubes are there from $1$ to $500$? How many are perfect squares among these cubes?

Answer 1

Let us find the perfect cubes between $1$ to $500$.

We know that, $1^3=1,2^3=8,3^3=27,4^3=64,5^3=125,6^3=216,7^3=343$ and $8^3=512$

This implies, taking number $\ge 8$ gives us cubes $\ge 500$. So, the rest numbers i.e. $8,9,10,....$ have their cubes greater than $500$.

So, only these $7$ numbers have their cubes between $1$ to $500$.

Thus, there are $7$ perfect cubes from $1$ to $500$.

Of these $64$ is the only perfect square $\because 64=4^3=8^2$.