Hello, I came across this problem and I wasn't entirely sure on how to do it. Seeing if someone can give me a hand with it? Any help is greatly appreciated, thanks!
Let's call $\displaystyle n_2\,,\,\,n_3$ to the number of squares (cubes) less than 1,000:
$\displaystyle 10^3=1,000\Longrightarrow n_3=10$
$\displaystyle 31^2=961\,,\,32^2=1,024>1,000\Longrightarrow n_3=31$.
Now, just take out ONCE the repeated ones: for example, $\displaystyle 64=4^3=8^2$, so we must erase 64 since it appears twice...
Tonio
Yes thats right.
An easy way to see this is all perfect squares are of the form $\displaystyle k^2$, all perfect cubes are of the form $\displaystyle k^3$ and all numbers that are both a perfect square and a perfect cube are $\displaystyle k^6$. Using inclusion exculsion, that means for any $\displaystyle N $ the number of numbers that are a perfect square or a perfect cube less than N are $\displaystyle \lfloor N^{\frac{1}{2}} \rfloor + \lfloor N^{\frac{1}{3}} \rfloor - \lfloor N^{\frac{1}{6}} \rfloor$