Question in proofs class. No idea how to figure this one out. I've been giving all questions an honest effort - this seems unorthodox.
Please help get me started
2026-03-25 14:23:56.1774448636
How many integers between 1 and 10,000 are neither squares nor cubes?
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Squares: $1^2,2^2,...,100^2 \Rightarrow$ we have 100 squares.
Cubes: $1^3,2^3,..., \lfloor \sqrt[3] {10000} \rfloor^3=21^3=9261 \Rightarrow$ we have 21 cubes.
Numbers that are squares and cubes: $1^6,2^6,..., \lfloor \sqrt[6] {10000} \rfloor^6=4^6=4096 \Rightarrow$ we have 4 numbers that are squares and cubes.
By inclusion-exclusion principle, follow that we have $10000-100-21+4=9883$ number which are neither squares nor cubes.