If $a = x^{m + n} . y^{l}; b = x^{n + l} . y^{m}$ and $c = x^{l + m} . y^{n}$ prove that : $a^{m - n} . b^{n - l} . c^{l - m} = 1$

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Solution

$a = x^{m + n} . y^{l}$
$a^{m - n} = (x^{m + n} . y^{l})^{m - n}$
$a^{m - n} = x^{m^{2} - n^{2}} . y^{m l - n l}$

$b = x^{n + l} . y^{m}$
$b^{n - l} = (x^{n + l} y^{m})^{n - l}$
$b^{n - l} = x^{n^{2} - l^{2}} . y^{m n - m l}$

$c = x^{l + m} . y^{n}$
$c^{l - m} = (x^{l + m} . y^{n})^{l - m}$
$c^{l - m} = x^{l^{2} - m^{2}} . y^{n l - m n}$

$a^{m - n} . b^{n - l} . c^{l - m} = x^{m^{2} - n^{2} + n^{2} - l^{2} + l^{2} - m^{2}} . y^{m l - n l + m n - m l + n l - m n}$
$= x^{0} . y^{0}$
$= 1$

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