Squaring a floor value

Question

I know this is a very crude question and will receive down votes but my question still remains:

Is it true that $$\lfloor{x}\rfloor^2 = \lfloor{x^2}\rfloor$$

Or generalising it to any arbitrary power $k$: $$\lfloor{x}\rfloor^k = \lfloor{x^k}\rfloor$$

Thanks a lot!

Answer 1

Hint : take $$x=\sqrt2+1$$ now see $$\lfloor x^2 \rfloor=\lfloor (\sqrt2+1)^2\rfloor =\\ \lfloor (2+2\sqrt2+1)\rfloor =\\3+\lfloor (2\sqrt2)\rfloor =5\\$$ vs $$(\lfloor x \rfloor)^2=\\ (\lfloor \sqrt2+1 \rfloor)^2=\\2^2=4$$ so $$\lfloor x^2 \rfloor \neq (\lfloor x \rfloor)^2$$it is better to say $$\lfloor x^2 \rfloor \geq (\lfloor x \rfloor)^2$$