Question

Cho M=\(\left(\frac{1}{\sqrt{x}-1}-\frac{1}{x\sqrt{x}-1}\right).\frac{3\sqrt{x}-3}{x+\sqrt{x}}\)

a, Rút gọn M

b, Tìm x để M có giá trị nguyên

Answer 1

\(Dk:x>0;x\ne1\)

\(M=\left(\frac{1}{\sqrt{x}-1}-\frac{1}{x\sqrt{x}-1}\right).\frac{3\sqrt{x}-3}{x+\sqrt{x}}=\left(\frac{x+\sqrt{x}+1}{\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}-\frac{1}{x\sqrt{x}-1}\right).\frac{3\left(\sqrt{x}-1\right)}{\sqrt{x}\left(\sqrt{x}+1\right)}=\frac{\sqrt{x}\left(\sqrt{x}+1\right)}{\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}.\frac{3\left(\sqrt{x}-1\right)}{\sqrt{x}\left(\sqrt{x}+1\right)}=\frac{3}{x+\sqrt{x}+1}\in Z\Rightarrow x+\sqrt{x}\in\left\{0;2\right\}\Rightarrow x\in\left\{0;1\right\}\Rightarrow x\in\varnothing\)