Simplify and express as positive indices:
(xy)m−n.(yz)n−l.(zx)l−m(xy)^{m-n}.(yz)^{n-l}.(zx)^{l-m}(xy)m−n.(yz)n−l.(zx)l−m
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(xy)m−n.(yz)n−l.(zx)l−m=(xm−nym−n).(yn−lzn−l).(zl−mxl−m)=x(m−n)+(l−m)y(m−n)+(n−l)z(n−l)+(l−m)=x−n+lym−lzn−m=xlymznxnylzm(xy)^{m-n}.(yz)^{n-l}.(zx)^{l-m}\\[1em] = (x^{m-n}y^{m-n}).(y^{n-l}z^{n-l}).(z^{l-m}x^{l-m})\\[1em] = x^{(m-n)+(l-m)}y^{(m-n)+(n-l)}z^{(n-l)+(l-m)}\\[1em] = x^{-n+l}y^{m-l}z^{n-m}\\[1em] = \dfrac{x^ly^mz^n}{x^ny^lz^m}(xy)m−n.(yz)n−l.(zx)l−m=(xm−nym−n).(yn−lzn−l).(zl−mxl−m)=x(m−n)+(l−m)y(m−n)+(n−l)z(n−l)+(l−m)=x−n+lym−lzn−m=xnylzmxlymzn
(xy)m−n.(yz)n−l.(zx)l−m=x−n+lym−lzn−m(xy)^{m-n}.(yz)^{n-l}.(zx)^{l-m} = x^{-n+l}y^{m-l}z^{n-m}(xy)m−n.(yz)n−l.(zx)l−m=x−n+lym−lzn−m
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[32x−5243y−5]−15\Big[\dfrac{32x^{-5}}{243y^{-5}}\Big]^{\dfrac{-1}{5}}[243y−532x−5]5−1
(a−2b)12×(ab−3)13(a^{-2}b)^{\dfrac{1}{2}} \times (ab^{-3})^{\dfrac{1}{3}}(a−2b)21×(ab−3)31
Show that: [xax−b]a−b.[xbx−c]b−c.[xcx−a]c−a=1\Big[\dfrac{x^a}{x^{-b}}\Big]^{a-b}.\Big[\dfrac{x^b}{x^{-c}}\Big]^{b-c}.\Big[\dfrac{x^c}{x^{-a}}\Big]^{c-a} = 1[x−bxa]a−b.[x−cxb]b−c.[x−axc]c−a=1
Evaluate: x5+n×(x2)3n+1x7n−2\dfrac{x^{5+n}\times(x^2)^{3n+1}}{x^{7n-2}}x7n−2x5+n×(x2)3n+1
