Question

\[ \sqrt[lm]{\frac{x^l}{x^m}} \cdot \sqrt[mn]{\frac{x^m}{x^n}} \cdot \sqrt[nl]{\frac{x^n}{x^l}} \]

Solution

\[ = \left(x^{l-m}\right)^{\frac{1}{lm}} \cdot \left(x^{m-n}\right)^{\frac{1}{mn}} \cdot \left(x^{n-l}\right)^{\frac{1}{nl}} \] \[ = x^{\frac{l-m}{lm}} \cdot x^{\frac{m-n}{mn}} \cdot x^{\frac{n-l}{nl}} \] \[ = x^{\left(\frac{l-m}{lm} + \frac{m-n}{mn} + \frac{n-l}{nl}\right)} \] \[ = x^{\left(\frac{1}{m}-\frac{1}{l} + \frac{1}{n}-\frac{1}{m} + \frac{1}{l}-\frac{1}{n}\right)} \] \[ = x^0 \] \[ = 1 \]

Answer

\[ \boxed{1} \]

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