<s>[INST] Approximate to whole number without calculator Approximate this number to the nearest whole number without a calculator:
$2^{\sqrt{5}}$
I don't know how to do this problem. Can you help me? My answer key says $5$.
Also, how do I approximate $2^\pi$ and $3^e$? The answers are $9$ and $20$, respectively.
 [/INST] i can get $4 < 2^{\sqrt 5} < 5.$  use the fact that $\sqrt x$ is concave down so the tangent approximation $\sqrt 5 = \sqrt {4 + 1)} < \sqrt 4 + {1 \over 2 \sqrt 4} = {5 \over 4}.$  so we have $2 < \sqrt 5 < {5 \over 4}$ and $2 = {512 \over 256} < {625 \over 256}$ gives us $2^{1/4} < {5 \over 4}.$
so far we have, $2 < \sqrt 5 < {5 \over 4}$ and $2 ^{1/4} < {5 \over 4}$ exponentiating shows $4 = 2^2 < 2^\sqrt 5  < 2^{5 \over 4} = 2^2 2^{1 \over 4} < 4 {5 \over 4} = 5.$  
</s>