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If $a^{m} . a^{n} = a^{m n}$ , then $m (n - 2) + n (m - 2)$ is

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$- \frac{m}{n}$

$m - n$

$- m n$

answer is A.

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Detailed Solution

$a^{m} . a^{n} = a^{m n}$

$a^{m} = \frac{a^{m n}}{a^{n}} = a^{m n - n} (∵ \frac{a^{m}}{a^{n}} = a^{m - n} w h e r e m > n)$

$m = m n - n (∵ a^{m} = a^{n} \Leftrightarrow m = n)$

$m + n = m n$

$∴ m (n - 2) + n (m - 2) = m n - 2 m + m n - 2 n$

$= 2 m n - 2 (m + n)$

$= 2 m n - 2 (m n)$

$= 2 m n - 2 m n = 0$

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