<s>[INST] Gauge fractions with exponents - No Calculator How does one (without the use of Calculator) determine that $5/6$ is less than $(35/36)^6$? How is this done mentally?
 [/INST] If you look at your second number, it can be written as 
$$( 1  -  \frac{1}{36} )^6 \ = \ 1 \ - \ 6 \cdot  1^5 \cdot \frac{1}{36} \ + \ \binom 62 \cdot 1^4 \cdot (\frac{1}{36})^2 \ - \ ...  $$
with the remaining (unwritten) terms being very small (the first two terms equal $ \ \frac{5}{6} \ $ ) .  So $ \ \frac{5}{6} \ $ ends up being slightly smaller than $ ( 1  -  \frac{1}{36} )^6 \ $ (by a little less than $ \ 15  \cdot \frac{1}{36^2} \ = \ \frac{5}{432} \ \approx  \ 0.0116 \ $ ) .
[This is what Daniel Fischer is briefly describing.]
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