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What is the nearest integer to (2+root3)^9?

 Sep 7, 2018
 #1
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I don't see any really clever way of doing this.  You can expand it using the binomial theorem to get

 

\(\left(2+\sqrt{3}\right)^9 = \sum \limits_{k=0}^9~\dbinom{9}{k}2^k~3^{(9-k)/2}\)

 

if you then expand all that out and combine things you'll have an expression that looks like

 

\(a + b \sqrt{3}\)

 

b above turns out to be 40545 so you'll need to take the square root of 3 out to 5 decimal places

 Sep 7, 2018
 #2
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70226 + 40545 sqrt(3) = 70,226 + 70,225.999992880 =~140,452

 Sep 8, 2018

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