(a) Điều kiện: \(\left\{{}\begin{matrix}x\ge0\\\sqrt{x}-3\ne0\end{matrix}\right.\) \(\Leftrightarrow\left\{{}\begin{matrix}x\ge0\\x\ne9\end{matrix}\right.\).
Ta có: \(\dfrac{\sqrt{x}-1}{\sqrt{x}-3}=\dfrac{\left(\sqrt{x}-3\right)+2}{\sqrt{x}-3}=1+\dfrac{2}{\sqrt{x}-3}\)
Biểu thức nguyên khi: \(\left(\sqrt{x}-3\right)\inƯ\left(2\right)=\left\{\pm1;\pm2\right\}\).
\(\Rightarrow\left[{}\begin{matrix}\sqrt{x}-3=1\\\sqrt{x}-3=-1\\\sqrt{x}-3=2\\\sqrt{x}-3=-2\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}x=16\left(TM\right)\\x=4\left(TM\right)\\x=25\left(TM\right)\\x=1\left(TM\right)\end{matrix}\right.\).
Vậy: \(x\in\left\{1;4;16;25\right\}.\)
(b) Điều kiện: \(\left\{{}\begin{matrix}x\ge0\\\sqrt{x}-3\ne0\end{matrix}\right.\) \(\Leftrightarrow\left\{{}\begin{matrix}x\ge0\\x\ne9\end{matrix}\right..\)
Ta có: \(\dfrac{3\sqrt{x}}{\sqrt{x}-3}=\dfrac{3\left(\sqrt{x}-3\right)-9}{\sqrt{x}-3}=3-\dfrac{9}{\sqrt{x}-3}\)
Biểu thức có giá trị nguyên khi: \(\left(\sqrt{x}-3\right)\inƯ\left(9\right)=\left\{\pm1;\pm3;\pm9\right\}\)
Mà: \(\sqrt{x}-3\ge-3\) (từ điều kiện \(x\ge0\))
\(\Rightarrow\left(\sqrt{x}-3\right)\in\left\{\pm1;\pm3;9\right\}\).
Giải ra từng trường hợp, ta được: \(x\in\left\{16;4;36;0;144\right\}\)
Kết hợp với điều kiện thì \(x\in\left\{0;4;16;36;144\right\}\).
a) \(\dfrac{\sqrt[]{x}-1}{\sqrt[]{x}-3}\in Z\left(x\ge0;x\ne9\right)\)
\(\Leftrightarrow\sqrt[]{x}-1⋮\sqrt[]{x}-3\)
\(\Leftrightarrow\sqrt[]{x}-1-\left(\sqrt[]{x}-3\right)⋮\sqrt[]{x}-3\)
\(\Leftrightarrow\sqrt[]{x}-1-\sqrt[]{x}+3⋮\sqrt[]{x}-3\)
\(\Leftrightarrow2⋮\sqrt[]{x}-3\)
\(\Leftrightarrow\sqrt[]{x}-3\in U\left(2\right)=\left\{-1;1;-2;2\right\}\)
\(\Leftrightarrow x\in\left\{4;16;1;25\right\}\)
b) \(\dfrac{3\sqrt[]{x}}{\sqrt[]{x}-3}\in Z\left(x\ge0;x\ne9\right)\)
\(\Leftrightarrow3\sqrt[]{x}⋮\sqrt[]{x}-3\)
\(\Leftrightarrow3\sqrt[]{x}-3\left(\sqrt[]{x}-3\right)⋮\sqrt[]{x}-3\)
\(\Leftrightarrow3\sqrt[]{x}-3\sqrt[]{x}+9⋮\sqrt[]{x}-3\)
\(\Leftrightarrow9⋮\sqrt[]{x}-3\)
\(\Leftrightarrow\sqrt[]{x}-3\in U\left(9\right)=\left\{-1;1-3;3;-9;9\right\}\)
\(\Leftrightarrow x\in\left\{4;160;36;\varnothing;144\right\}\)
\(\Leftrightarrow x\in\left\{4;160;36;144\right\}\)
