Question

Without a calculator how would you solve this?

Answer 1

the largest number that is less than (2+√3)^6 is 2701

Explanation:

`\bigstar \ (2+√(3))^6= \\\\C{}^(0)_(6)2^(6)+C{}^(1)_(6)2^(5)√(3)^1+C{}^(2)_(6)2^(4)√(3)^2+C{}^(3)_(6)2^(3)√(3)^3+C{}^(4)_(6)2^(2)√(3)^4+C{}^(5)_(6)2^(1)√(3)^5+C{}^(6)_(6)√(3)^6`

`=64+192√(3)+720 +480√(3) +540+108√(3) +27\\=1351+780√(3)`

`\bigstar\ (2-√(3) )^6 =\\\\1351-780√(3)`

`\bigstar\ \bigstar\ (2+√(3) )^6+(2-√(3) )^6 =\\\\1351+780√(3)+1351-780√(3)`

`\Longrightarrow(2+√(3) )^6+(2-√(3) )^6 =2702`

`\Longrightarrow(2+√(3) )^6 =2702-(2-√(3) )^6`

`0 < \left( 2-√(3) \right) < 1 \Longrightarrow 0 < \left( 2-√(3) \right)^(6) < 1 \Longrightarrow-1 < \left( 2-√(3) \right)^(6) < 0`

`\Longrightarrow2702-1 < 2702-\left( 2-√(3) \right)^(6) < 2702+0`

`\Longrightarrow 2701 < \left( 2+√(3) \right)^(6) < 2702`