<s>[INST] Comparing numbers of the form $c+\sqrt{b}$ (eg, $3+3\sqrt{3}$ and $4+2\sqrt{5}$) without a calculator It is easy to compare to numbers of the form $a\sqrt{b}$, simply by comparing their squares, for example $3\sqrt{3}$ and $2\sqrt{5}$.
But what if we have $a=3+3\sqrt{3}$ and $b=4+2\sqrt{5}$ for example?
How to compare them without using a calculator? Is there a method that works for all numbers of the form $c+\sqrt{b}$?
I found a similar question to this on the site, but that one is a bit different and I am looking for an answer suitable for elementary students to understand. (but I would love to know any kind of answer even if it does not fit the elementary level).
I was thinking about subtracting $b-a$ and checking wether it is negative or positive,
I got $b-a=1+2\sqrt{5}-3\sqrt{3}$ but $2\sqrt{5}-3\sqrt{3}<0$
(I don't want to continue from here by approximating $\sqrt{5}=2.something$, because we will be somehow using the calculator in our mind; so I am stuck here)
 [/INST] You can use binomial theorem result
$(1\pm x)^n\approx (1\pm nx)$ provided that $x<<1$
Therefore you have $3+3\sqrt3=3+3(4-1)^{\frac{1}{2}}\approx3+6(1-0.25)^{\frac{1}{2}}= 3+6(1-0.125)=9-0.75=8.25$
Similarly  $4+2\sqrt5=4+2(4+1)^{\frac{1}{2}}\approx4+4(1+0.25)^{\frac{1}{2}}= 4+4(1+0.125)=8+0.5=8.5$
This would work better when you have large numbers involved inside radical
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