Simplify the following:
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
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Given,
⇒xlxmlm.xmxnmn.xnxlnl=xl−mlm.xm−nmn.xn−lnl=(x)l−mlm.(x)m−nmn.(x)n−lnl=(x)l−mlm+m−nmn+n−lnl=(x)(l−m)n+(m−n)l+(n−l)mlmn=(x)ln−mn+ml−nl+nm−lmlmn=(x)0=1.\Rightarrow \sqrt[lm]{\dfrac{x^l}{x^m}}.\sqrt[mn]{\dfrac{x^m}{x^n}}.\sqrt[nl]{\dfrac{x^n}{x^l}} \\[1em] = \sqrt[lm]{x^{l - m}}.\sqrt[mn]{x^{m - n}}.\sqrt[nl]{x^{n - l}} \\[1em] = (x)^{\dfrac{l - m}{lm}}.(x )^{\dfrac{m - n}{mn}}.(x)^{\dfrac{n - l}{nl}} \\[1em] = (x)^{\dfrac{l - m}{lm} + \dfrac{m - n}{mn} + \dfrac{n - l}{nl}} \\[1em] = (x)^{\dfrac{(l - m)n + (m - n)l + (n - l)m}{lmn}} \\[1em] = (x)^{\dfrac{ln - mn + ml - nl + nm - lm}{lmn}} \\[1em] = (x)^0 \\[1em] = 1.⇒lmxmxl.mnxnxm.nlxlxn=lmxl−m.mnxm−n.nlxn−l=(x)lml−m.(x)mnm−n.(x)nln−l=(x)lml−m+mnm−n+nln−l=(x)lmn(l−m)n+(m−n)l+(n−l)m=(x)lmnln−mn+ml−nl+nm−lm=(x)0=1.
Hence, 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 = 1.
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(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
(xa+bxc)a−b.(xb+cxa)b−c.(xc+axb)c−a\Big(\dfrac{x^{a + b}}{x^c}\Big)^{a - b}.\Big(\dfrac{x^{b + c}}{x^a}\Big)^{b - c}.\Big(\dfrac{x^{c + a}}{x^b}\Big)^{c - a}(xcxa+b)a−b.(xaxb+c)b−c.(xbxc+a)c−a
(xaxb)a2+ab+b2.(xbxc)b2+bc+c2.(xcxa)c2+ac+a2\Big(\dfrac{x^a}{x^b}\Big)^{a^2 + ab + b^2}.\Big(\dfrac{x^b}{x^c}\Big)^{b^2 + bc + c^2}.\Big(\dfrac{x^c}{x^a}\Big)^{c^2 + ac + a^2}(xbxa)a2+ab+b2.(xcxb)b2+bc+c2.(xaxc)c2+ac+a2
(xax−b)a2−ab+b2.(xbx−c)b2−bc+c2.(xcx−a)c2−ca+a2\Big(\dfrac{x^a}{x^{-b}}\Big)^{a^2 - ab + b^2}.\Big(\dfrac{x^b}{x^{-c}}\Big)^{b^2 - bc + c^2}.\Big(\dfrac{x^c}{x^{-a}}\Big)^{c^2 - ca + a^2}(x−bxa)a2−ab+b2.(x−cxb)b2−bc+c2.(x−axc)c2−ca+a2
