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The square root of 1,000,000 is 1000; therefore, there will be 1000 perfect squares:

1²  =  1

2²  =  4

3²  =  9

...

998²  =  996,004

999²  =  998,001

1000²  =  1,000,000

 

Now, we have to cross-check that with a list of perfect cubes.

Since the cube root of 1,000,000 is 100, there will be 100 numbers on this list.

1³  =  1                                 Cross this one off

2³  =  8

3³  =  27

4³  =  64                              Cross this one off  

...

98³  =  941,192

99³  =  970,299

100³  =  1,000,000               Cross this one off

 

So, there will be a total of 1000 from the list of perfect squares plus all those on the list of perfect cubes that 

aren't crossed off.

 

Then, subtract that total from 1,000,000.