| Find the value of \[m-n,\]if \[\frac{{{9}^{n}}\times {{3}^{2}}\times {{\left( {{3}^{-\frac{n}{2}}} \right)}^{-2}}-{{(27)}^{n}}}{{{3}^{3m}}\times {{2}^{3}}}=\frac{1}{27}\] |
A) \[1\]
B) \[-\,\,2\]
C) \[-\,\,1\]
D) \[2\]
Correct Answer: A
Solution :
| \[\frac{{{9}^{n}}\times {{3}^{2}}\times {{({{3}^{-n/2}})}^{-2}}-{{(27)}^{n}}}{{{3}^{3m}}\times {{2}^{3}}}=\frac{1}{27}\] |
| \[\Rightarrow \]\[\frac{{{3}^{2n}}\times {{3}^{2}}\times {{3}^{n}}\times {{3}^{3n}}}{{{3}^{3m}}\times {{2}^{3}}}=\frac{1}{27}\]\[\Rightarrow \]\[\frac{{{3}^{3n+2}}-{{3}^{3n}}}{{{3}^{3m}}\times 8}=\frac{1}{27}\] |
| \[\Rightarrow \] \[\frac{{{3}^{3n}}\times 8}{{{3}^{3m}}\times 8}=\frac{1}{{{(3)}^{3}}}\]\[\Rightarrow \]\[{{\{{{(3)}^{3}}\}}^{n-\,\,m}}={{3}^{-\,\,3}}\] |
| \[\Rightarrow \] \[n-m=-1\]or \[m-n=1\] |
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