Simplify the following:
(xmxn)l.(xnxl)m.(xlxm)n\Big(\dfrac{x^m}{x^n}\Big)^l.\Big(\dfrac{x^n}{x^l}\Big)^m.\Big(\dfrac{x^l}{x^m}\Big)^n(xnxm)l.(xlxn)m.(xmxl)n
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Given,
⇒(xmxn)l.(xnxl)m.(xlxm)n=(xm.x−n)l.(xn.x−l)m.(xl.x−m)n=(xm−n)l.(xn−l)m.(xl−m)n=xml−nl.xnm−lm.xln−mn=xml−nl+nm−lm+ln−mn=x0=1.\Rightarrow \Big(\dfrac{x^m}{x^n}\Big)^l.\Big(\dfrac{x^n}{x^l}\Big)^m.\Big(\dfrac{x^l}{x^m}\Big)^n = (x^m.x^{-n})^l.(x^n.x^{-l})^m.(x^l.x^{-m})^n \\[1em] = (x^{m - n})^l.(x^{n - l})^m.(x^{l - m})^n \\[1em] = x^{ml - nl}.x^{nm - lm}.x^{ln - mn} \\[1em] = x^{ml - nl + nm - lm + ln - mn} \\[1em] = x^0 = 1.⇒(xnxm)l.(xlxn)m.(xmxl)n=(xm.x−n)l.(xn.x−l)m.(xl.x−m)n=(xm−n)l.(xn−l)m.(xl−m)n=xml−nl.xnm−lm.xln−mn=xml−nl+nm−lm+ln−mn=x0=1.
Hence, (xmxn)l.(xnxl)m.(xlxm)n\Big(\dfrac{x^m}{x^n}\Big)^l.\Big(\dfrac{x^n}{x^l}\Big)^m.\Big(\dfrac{x^l}{x^m}\Big)^n(xnxm)l.(xlxn)m.(xmxl)n = 1.
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xlxmlm.xmxnmn.xnxlnl\sqrt[lm]{\dfrac{x^l}{x^m}}.\sqrt[mn]{\dfrac{x^m}{x^n}}.\sqrt[nl]{\dfrac{x^n}{x^l}}lmxmxl.mnxnxm.nlxlxn
