Câu hỏi của Trương Thanh Nhi

Question

$\left(2+\sqrt{3}\right)\left(2+\sqrt{2+\sqrt{3}}\right)\left(2-\sqrt{2+\sqrt{3}}\right)$

Nguyễn Lê Phước Thịnh · Accepted Answer

Ta có: $\left(2+\sqrt{3}\right)\cdot\left(2+\sqrt{2+\sqrt{3}}\right)\cdot\left(2-\sqrt{2+\sqrt{3}}\right)$

$=\left(2+\sqrt{3}\right)\cdot\left[2^2-\sqrt{\left(2+\sqrt{3}\right)^2}\right]$

$=\left(2+\sqrt{3}\right)\cdot\left(4-\left|2+\sqrt{3}\right|\right)$

$=\left(2+\sqrt{3}\right)\cdot\left[4-\left(2+\sqrt{3}\right)\right]$(Vì $2>\sqrt{3}>0$)

$=\left(2+\sqrt{3}\right)\cdot\left(4-2-\sqrt{3}\right)$

$=\left(2+\sqrt{3}\right)\cdot\left(2-\sqrt{3}\right)$

$=4-3=1$Câu trả lời: Ta có: $\left(2+\sqrt{3}\right)\cdot\left(2+\sqrt{2+\sqrt{3}}\right)\cdot\left(2-\sqrt{2+\sqrt{3}}\right)$

$=\left(2+\sqrt{3}\right)\cdot\left[2^2-\sqrt{\left(2+\sqrt{3}\right)^2}\right]$

$=\left(2+\sqrt{3}\right)\cdot\left(4-\left|2+\sqrt{3}\right|\right)$

$=\left(2+\sqrt{3}\right)\cdot\left[4-\left(2+\sqrt{3}\right)\right]$(Vì $2>\sqrt{3}>0$)

$=\left(2+\sqrt{3}\right)\cdot\left(4-2-\sqrt{3}\right)$

$=\left(2+\sqrt{3}\right)\cdot\left(2-\sqrt{3}\right)$

$=4-3=1$

Violympic toán 9