Question

prove m-n =2. need this by tomorrow

Answer 1

☺️HEY MATE! HERE'S YOUR ANSWER!☺️

$\frac{ {9}^{n + 1} \times ( {3}^{ { \frac{ - n}{2}) }^{ - 2} } - {27}^{n} }{( {3}^{m} \times {2})^{3} } = \frac{1}{729}$

$= > \frac{ {3}^{2(n + 1) \times ( {3})^{ \frac{ - n}{2} \times - 2 } - {(3)}^{3n} } }{( {3})^{3m} \times ( {2)}^{3} } = \frac{1}{729}$

$= > \frac{ {3}^{2n + 2} \times {3}^{n} - {3}^{3} }{ {3}^{m} \times {2}^{3} }= \frac{1}{ ({3})^{6} }$

$= > \frac{ {3}^{2n + 2 + n} - {3}^{3n} }{ {3}^{3m} \times 8 } = \frac{1}{( {3)}^{6} }$

$= > \frac{ {3}^{3n + 2} - {3}^{3n} }{ {3}^{m} \times 8 } = {3}^{ - 6}$

$= > \frac{ {3}^{3n} \times {3}^{2} - {3}^{3n} }{ {3}^{m} \times 8} = ( {3})^{ - 6}$

$= > \frac{ {3}^{n} ( {3}^{2} - 1) }{ {3}^{m} \times 8} = {3}^{ - 6}$

$= > \frac{ {3}^{3n - 3m} \times (9 - 1)}{8} = {3}^{ - 6}$

$= > \frac{ {3}^{3(n - m)} \times 8 }{8} = {3}^{ - 6}$

$= > {3}^{3(n - m)} = ( {3})^{ - 6}$

$= > 3(n - m) = - 6$

$= > n - m = \frac{ - 6}{3}$

$= > n - m = - 2$

$= > m - n = 2$

hence, proved✔️✔️✔️✔️✔️

hope it is helpful ✌️!