Find the greatest integer less than $10.3^3$ without using a calculator.

Write 10.3 as (10 + 0.3) then use the binomial expansion on (10 + 0.3)^3 to get:

10^3 + 3*10^2*0.3 + 3*10*0.3^2 + 0.3^3 or

1000 + 90 + 2.7 + 0.027 → 1092.727

Therefore the greatest integer less than 10.3^3 is 1092

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I would not have thought to do it that way.  Thanks Alan