Question
Simplify:$\sqrt[\text{lm}]{\frac{\text{x}^\text{l}}{\text{x}^\text{m}}}\times\sqrt[\text{mn}]{\frac{\text{x}^\text{m}}{\text{x}^\text{n}}}\times\sqrt[\text{nl}]{\frac{\text{x}^\text{n}}{\text{x}^\text{l}}}$
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Answer
$\sqrt[\text{lm}]{\frac{\text{x}^\text{l}}{\text{x}^\text{m}}}\times\sqrt[\text{mn}]{\frac{\text{x}^\text{m}}{\text{x}^\text{n}}}\times\sqrt[\text{nl}]{\frac{\text{x}^\text{n}}{\text{x}^\text{l}}}$$=\Big(\frac{\text{x}^{\text{l}}}{\text{x}^\text{m}}\Big)^{\frac{1}{\text{lm}}}\times\Big(\frac{\text{x}^\text{m}}{\text{x}^\text{n}}\Big)^{\frac{1}{\text{mn}}}\times\Big(\frac{\text{x}^\text{n}}{\text{x}^\text{l}}\Big)^{\frac{1}{\text{nl}}}$
$=\big(\text{x}^{\text{l}-\text{m}}\big)^{\frac{1}{\text{lm}}}\times\big(\text{x}^{\text{m}-\text{n}}\big)^{\frac{1}{\text{mn}}}\times\big(\text{x}^{\text{n}-\text{}1}\big)^{\frac{1}{\text{nl}}}$
$=\text{x}^{\frac{\text{l}-\text{m}}{\text{mn}}}\times\text{x}^{\frac{\text{m}-\text{n}}{\text{mn}}}\times\text{x}^{\frac{\text{n}-\text{l}}{\text{nl}}}$
$=\text{x}^{\frac{\text{l}-\text{m}}{\text{mn}}+\frac{\text{m}-\text{n}}{\text{mn}}+\frac{\text{n}-\text{l}}{\text{nl}}}$
$=\text{x}^{\frac{\text{n}(\text{l}-\text{m})+\text{l}(\text{m}-\text{n})+\text{m}(\text{n}-\text{l})}{\text{lmn}}}$
$=\text{x}^{\frac{\text{n}\text{l}-\text{m}+\text{l}\text{m}-\text{n}\text{l}+\text{m}\text{n}-\text{lm}}{\text{lmn}}}$
$=\text{x}^0$
$=\text{1}$
$=\big(\text{x}^{\text{l}-\text{m}}\big)^{\frac{1}{\text{lm}}}\times\big(\text{x}^{\text{m}-\text{n}}\big)^{\frac{1}{\text{mn}}}\times\big(\text{x}^{\text{n}-\text{}1}\big)^{\frac{1}{\text{nl}}}$
$=\text{x}^{\frac{\text{l}-\text{m}}{\text{mn}}}\times\text{x}^{\frac{\text{m}-\text{n}}{\text{mn}}}\times\text{x}^{\frac{\text{n}-\text{l}}{\text{nl}}}$
$=\text{x}^{\frac{\text{l}-\text{m}}{\text{mn}}+\frac{\text{m}-\text{n}}{\text{mn}}+\frac{\text{n}-\text{l}}{\text{nl}}}$
$=\text{x}^{\frac{\text{n}(\text{l}-\text{m})+\text{l}(\text{m}-\text{n})+\text{m}(\text{n}-\text{l})}{\text{lmn}}}$
$=\text{x}^{\frac{\text{n}\text{l}-\text{m}+\text{l}\text{m}-\text{n}\text{l}+\text{m}\text{n}-\text{lm}}{\text{lmn}}}$
$=\text{x}^0$
$=\text{1}$
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