Question

The greatest integer less than or equal to $(2+\sqrt{3}{)}^6$ is

• A. $2702$
• B. $2701$
• C. $1351$
• D. $1350$

Answer 1

The correct option is B $2701$

We have,
$(2+\sqrt{3}{)}^6+(2-\sqrt{3}{)}^6$
$=\left[2^6{+}^6{C}_1\right(2^5\left)\right(\sqrt{3})+...{+}^6{C}_5(2\left)\right(\sqrt{3}{)}^5)+(\sqrt{3}{)}^6]+\left[2^6{-}^6{C}_1\right(2^5\left)\right(\sqrt{3})+...{-}^6{C}_5(2\left)\right(\sqrt{3}{)}^5)+(\sqrt{3}{)}^6\left)\right]$
$=2[2^6+15\times (16\left)\right(3)+15(4\left)\right(9)+27]$
$=2702$
We have, $0<(2-\sqrt{3})<1$.
Hence, $0<(2-\sqrt{3}{)}^6<1$.
Hence, $2701<(2+\sqrt{3}{)}^6)<2702$.
Hence, option B is the correct answer.