Question

\[ a=x^{m+n}y^{l},\quad b=x^{n+l}y^{m},\quad c=x^{l+m}y^{n} \] \[ \text{Prove } a^{m-n}b^{n-l}c^{l-m}=1 \]

Solution

\[ a^{m-n} = x^{(m+n)(m-n)} y^{l(m-n)} \] \[ b^{n-l} = x^{(n+l)(n-l)} y^{m(n-l)} \] \[ c^{l-m} = x^{(l+m)(l-m)} y^{n(l-m)} \] \[ \Rightarrow a^{m-n}b^{n-l}c^{l-m} = x^{(m+n)(m-n)+(n+l)(n-l)+(l+m)(l-m)} \cdot y^{l(m-n)+m(n-l)+n(l-m)} \] \[ = x^{(m^2-n^2)+(n^2-l^2)+(l^2-m^2)} \cdot y^{(lm-ln+mn-ml+nl-nm)} \] \[ = x^0 \cdot y^0 \] \[ = 1 \]

Answer

\[ \boxed{1} \]

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