Question

[Tuấn Huy Trương]

a) Cho hai số thực dương a,b thỏa mãn a²+ b = 2ab. CMR: 3a² + b ≥ \(\dfrac{8}{3}\).

b) CMR với mọi số tự nhiên n thì : A = 2025ⁿ + 112ⁿ - 89ⁿ - 24ⁿ ⋮ 2024

c) P = \(\dfrac{\sqrt{x^3}-\sqrt{y^3}}{x-y}\)- \(\dfrac{x}{\sqrt{x}+\sqrt{y}}\) - \(\dfrac{y}{\sqrt{y}-\sqrt{x}}\). Rút gọn biểu thức P biết y>x>0

Answer 1

[Nguyễn Lê Phước Thịnh]

c: \(P=\frac{\sqrt{x^3}-\sqrt{y^3}}{x-y}-\frac{x}{\sqrt{x}+\sqrt{y}}-\frac{y}{\sqrt{y}-\sqrt{x}}\)

\(=\frac{x\cdot\sqrt{x}-y\sqrt{y}}{x-y}-\frac{x}{\sqrt{x}+\sqrt{y}}+\frac{y}{\sqrt{x}-\sqrt{y}}\)

\(=\frac{\left(x\sqrt{x}-y\sqrt{y}\right)-x\left(\sqrt{x}-\sqrt{y}\right)+y\left(\sqrt{x}+\sqrt{y}\right)}{\left(\sqrt{x}-\sqrt{y}\right)\left(\sqrt{x}+\sqrt{y}\right)}\)

\(=\frac{x\sqrt{x}-y\sqrt{y}-x\sqrt{x}+x\sqrt{y}+y\sqrt{x}+y\sqrt{y}}{x-y}=\frac{x\sqrt{y}+y\cdot\sqrt{x}}{\left(\sqrt{x}-\sqrt{y}\right)\left(\sqrt{x}+\sqrt{y}\right)}\)

\(=\frac{\sqrt{xy}\left(\sqrt{x}+\sqrt{y}\right)}{\left(\sqrt{x}-\sqrt{y}\right)\left(\sqrt{x}+\sqrt{y}\right)}=\frac{\sqrt{xy}}{\sqrt{x}-\sqrt{y}}\)