Question

Cho A:  \(\dfrac{x\sqrt{x}+5\sqrt{x}-12}{x-\sqrt{x}-6}-\dfrac{2\sqrt{x}+6}{\sqrt{x}+2}+\dfrac{\sqrt{X+3}}{3-\sqrt{x}}\)
a) Rút gọn A
b) Tính giá trị của bt A khi x=\(\dfrac{2\sqrt{1+\dfrac{\sqrt{3}}{2}}\left(\sqrt{3}-1\right)}{\sqrt{11+4\sqrt{7}-\sqrt{7}}}\)
c) Tìm GTNN của A

Answer 1

a: \(A=\dfrac{x\sqrt{x}+5\sqrt{x}-12-2\left(x-9\right)-\left(x+5\sqrt{x}+6\right)}{\left(\sqrt{x}+2\right)\left(\sqrt{x}-3\right)}\)

\(=\dfrac{x\sqrt{x}+5\sqrt{x}-12-2x+18-x-5\sqrt{x}-6}{\left(\sqrt{x}+2\right)\left(\sqrt{x}-3\right)}\)

\(=\dfrac{x\sqrt{x}-3\sqrt{x}}{\left(\sqrt{x}+2\right)\left(\sqrt{x}-3\right)}=\dfrac{x}{\sqrt{x}+2}\)

b: \(A=\dfrac{2\sqrt{\dfrac{2+\sqrt{3}}{\sqrt{2}}}\cdot\left(\sqrt{3}-1\right)}{\sqrt{2}}\)

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

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

\(=\sqrt{\dfrac{4\sqrt{2}}{2}}=\sqrt{2\sqrt{2}}\)

Khi \(x=\sqrt{2\sqrt{2}}\) vào A, ta được:

\(A=\dfrac{\sqrt{2\sqrt{2}}}{\sqrt{\sqrt{2\sqrt{2}}}+2}\)