This method will work for 1 followed by any number of zeros;
Let's take 1000 for example.
1000 has three zeros, and 3 is divisible by 3, so 1000 is a perfect cube
The prime factorisation of 1000 will give us 2*2*2*5*5*5. Since this is cube roots, we group the 2's and 5's to get 2*5=10, and 10 is the cube root of 1000.
Next example, 10000, which has 5 zeros. 5 can't be divided by 3 to get equal whole numbers, so 10000 is not going to be a perfect cube number.
the prime factorisation of 10000 will give us 2*2*2*2*2*5*5*13. We can't group the 5's and the 13, so this shows that 10000 is not a perfect cube number.
This method can work for only a 1 followed by how many zeros it has. Pretty much, if the number of zeros can be divisible by 3 to get equal whole numbers, it's a perfect cube number!
Something else you might want to know,
Take 89 cubed
89 cubed - 89*89*89 = 684909
The number of digits of the cube are equal to the number of 89's that are their, which is 3 89's. This shows that the number of digits that are there for the extended form of 89 cubed, ( which is 6 here) gives us how many digits there will be in the answer, which is 6. This is for odd numbers, but for even numbers it would change.
Now let's take 34
34 cubed= 34*34*34. The extended form has 6 digits. But for this, we have to actually minus 1 form the total number of digits. 6-1=5. So, the answer should have 5 digits. Let's check. The cube of 34 gives us( drumroll please) 39304, which has ( drumroll) 5 digits! This method works for whole number cubing.
Sorry if this is kind of complicated, and again if this does not satisfy you.
By the way, I also had the same question, and did some research on the internet. I came up with nothing! The stuff I gave you above is pure observation and multiple testings! If you're not sure if they are not correct, try out those methods on some numbers and see if they are actually accurate ( I've already tried it, that's why I'm posting this here, duh)! Have fun!