Question

Simplify (2+√3)² + 1/(2+√3)²​

Answer 1

Explanation:

$(2+\sqrt3)^2 = 2^2 + 2(2)(\sqrt3) + (\sqrt3)^2=4 + 4\sqrt3+ 3$   [∵ $(a+b)^2=a^2 + 2ab + b^2$]

∴ $(2+\sqrt3)^2 = 7 + 4\sqrt3$

∴ $(2+\sqrt3)^2+\frac{1}{(2+\sqrt3)^2}$

$= (7+4\sqrt3)+\frac{1}{7+4\sqrt3}$

$= (7+4\sqrt3)+ \frac{(7-4\sqrt3)}{(7+4\sqrt3)(7-4\sqrt3)}$

$=(7+4\sqrt3)+\frac{(7-4\sqrt3)}{7^2 - (4\sqrt3)^2}$ [∵ $(a+b)(a-b)=a^2 - b^2$]

$=(7=4\sqrt3)+ \frac{(7-4\sqrt3)}{49-(16)(3)}=(7=4\sqrt3)+ \frac{(7-4\sqrt3)}{49-48}$

$=(7+4\sqrt3)+(7-4\sqrt3)$

$= 14$

Answer: 14

I hope this helps. :D