\(\begin{array}{l}
1)\quad \sqrt{x\sqrt[3]{x^2\sqrt[k]{x^3}}} = x^{\tfrac{23}{24}}\qquad (x>0;k\ne 0)\\
\Leftrightarrow \sqrt{x\sqrt[3]{x^2.x^{\tfrac3k}}} = x^{\tfrac{23}{24}}\\
\Leftrightarrow \sqrt{x\sqrt[3]{x^{2 + \tfrac3k}}} = x^{\tfrac{23}{24}}\\
\Leftrightarrow \sqrt{x.x^{\tfrac23 + \tfrac1k}} = x^{\tfrac{23}{24}}\\
\Leftrightarrow \sqrt{x^{\tfrac53 + \tfrac1k}} = x^{\tfrac{23}{24}}\\
\Leftrightarrow x^{\tfrac56 + \tfrac{1}{2k}} = x^{\tfrac{23}{24}}\qquad (*)\\
+)\quad \text{Với}\ x = 1\ \text{ta được:}\\
(*)\Leftrightarrow 1^{\tfrac56 + \tfrac{1}{2k}} = 1^{\tfrac{23}{24}}\quad \left(\text{luôn đúng}\ \ \forall k\ne 0\right)\\
\Rightarrow k\in \Bbb R\backslash \{0\}\\
+)\quad \text{Với}\ x \ne 1\ \text{ta được:}\\
(*) \Leftrightarrow \dfrac56 + \dfrac{1}{2k} = \dfrac{23}{24}\\
\Leftrightarrow \dfrac{1}{2k} = \dfrac{1}{8}\\
\Leftrightarrow k = 4\\
2)\quad \sqrt[3]{\dfrac ab\cdot \sqrt[4]{\dfrac ba\cdot \sqrt{\dfrac ab}}} = \left(\dfrac ab\right)^m\qquad (a;b>0)\\
\Leftrightarrow \sqrt[3]{\dfrac ab\cdot \sqrt[4]{\left(\dfrac ab\right)^{-1}\cdot \left(\dfrac ab\right)^\tfrac12}} = \left(\dfrac ab\right)^m\\
\Leftrightarrow \sqrt[3]{\dfrac ab\cdot \sqrt[4]{\left(\dfrac ab\right)^{-\tfrac12}}} = \left(\dfrac ab\right)^m\\
\Leftrightarrow \sqrt[3]{\dfrac ab\cdot\left(\dfrac ab\right)^{-\tfrac18}}=\left(\dfrac ab\right)^m\\
\Leftrightarrow \sqrt[3]{\left(\dfrac ab\right)^{\tfrac78}} = \left(\dfrac ab\right)^m\\
\Leftrightarrow \left(\dfrac ab\right)^{\tfrac{7}{24}} = \left(\dfrac ab\right)^m\qquad (**)\\
+)\quad \text{Với}\ a = b >0\ \text{ta được:}\\
(**) \Leftrightarrow 1^{\tfrac{7}{24}} = 1^m\\
\Leftrightarrow m \in \Bbb R\\
+)\quad \text{Với}\ a \ne b\ \text{ta được:}\\
(**) \Leftrightarrow m = \dfrac{7}{24}\\
\end{array}\)
