a) \(\dfrac{\sqrt[]{x}}{6\sqrt[]{x}+1}\in Z\left(x\ge0;x\in N\right)\)
\(\Leftrightarrow\sqrt[]{x}⋮6\sqrt[]{x}+1\)
\(\Leftrightarrow6\sqrt[]{x}+1-6\sqrt[]{x}⋮6\sqrt[]{x}+1\)
\(\Leftrightarrow1⋮6\sqrt[]{x}+1\)
\(\Leftrightarrow6\sqrt[]{x}+1\in U\left(1\right)=\left\{1\right\}\)
\(\Leftrightarrow6\sqrt[]{x}\in\left\{0\right\}\)
\(\Leftrightarrow x\in\left\{0\right\}\)
b) \(\dfrac{\sqrt[]{x}-2}{\sqrt[]{x}-3}\in Z\left(x\ge0;x\in N\right)\)
\(\Leftrightarrow\sqrt[]{x}-2⋮\sqrt[]{x}-3\)
\(\Leftrightarrow\sqrt[]{x}-2-\left(\sqrt[]{x}-3\right)⋮\sqrt[]{x}-3\)
\(\Leftrightarrow\sqrt[]{x}-2-\sqrt[]{x}+3⋮\sqrt[]{x}-3\)
\(\Leftrightarrow1⋮\sqrt[]{x}-3\)
\(\Leftrightarrow\sqrt[]{x}-3\in U\left(1\right)=\left\{1\right\}\)
\(\Leftrightarrow\sqrt[]{x}\in\left\{4\right\}\)
\(\Leftrightarrow x\in\left\{16\right\}\)
