The greatest integer less than or equal to (s | vedclass

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Question

(C) Let $(2+\sqrt{3})^5 = I + f$,where $I$ is an integer and $0 < f < 1$. Consider the expression $(2-\sqrt{3})^5 = f'$. Since $0 < 2-\sqrt{3} < 1$,it

Vedclass · Accepted Answer

$723$

Vedclass · Answer

(C) Let $(2+\sqrt{3})^5 = I + f$,where $I$ is an integer and $0 Consider the expression $(2-\sqrt{3})^5 = f'$. Since $0 Now,consider the sum $S = (2+\sqrt{3})^5 + (2-\sqrt{3})^5$. Using the binomial expansion,$S = 2 	imes [^5C_0 2^5 + ^5C_2 2^3 (\sqrt{3})^2 + ^5C_4 2^1 (\sqrt{3})^4]$. $S = 2 	imes [32 + 10 	imes 8 	imes 3 + 5 	imes 2 	imes 9] = 2 	imes [32 + 240 + 90] = 2 	imes 362 = 724$. Since $I + f + f' = 724$ and $0 Given $0 Therefore,$I + 1 = 724$,which implies $I = 723$.

The greatest integer less than or equal to $(\sqrt{3}+2)^5$ is

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