<s>[INST] Find the greatest integer in the expression below as a function of the given conditions Find the greatest integer less than $\sqrt{2^{100}+10^{10}}.$
Answer: $2^{50}$
I tried, but I can't finish:
$\sqrt{2^{100}+(2\cdot5)^{10}}=\sqrt{2^{10}(2^{90}+5^{10})}=2^5\sqrt{2^{90}+5^{10}}$
 [/INST] It results that
$2^{50}=\sqrt{2^{100}}<\sqrt{2^{100}+10^{10}}<\sqrt{2^{100}+16^{10}}=$
$=\sqrt{2^{100}+2^{40}}<\sqrt{2^{100}+2^{51}+1}=\sqrt{\left(2^{50}+1\right)^2}=2^{50}+1$
Hence ,
$2^{50}<\sqrt{2^{100}+{10^{10}}}<2^{50}+1\;.$
Consequently,
$\left\lfloor\sqrt{2^{100}+10^{10}}\right\rfloor=2^{50}\;.$
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