Question

\[ x=a^{m+n},\quad y=a^{n+l},\quad z=a^{l+m} \] \[ \text{Prove } x^m y^n z^l = x^n y^l z^m \]

Solution

\[ x^m y^n z^l = a^{m(m+n)} \cdot a^{n(n+l)} \cdot a^{l(l+m)} \] \[ = a^{m^2+mn+n^2+nl+l^2+lm} \] \[ x^n y^l z^m = a^{n(m+n)} \cdot a^{l(n+l)} \cdot a^{m(l+m)} \] \[ = a^{mn+n^2+nl+l^2+lm+m^2} \] \[ \Rightarrow x^m y^n z^l = x^n y^l z^m \]

Answer

\[ \boxed{x^m y^n z^l = x^n y^l z^m} \]

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