question_answer

Find the value of x for which     \[{{\left( \frac{4}{9} \right)}^{4}}\times {{\left( \frac{4}{9} \right)}^{-7}}={{\left( \frac{4}{9} \right)}^{2x-1}}\]

Answer:

Since, \[{{\left( \frac{4}{9} \right)}^{4}}\times {{\left( \frac{4}{9} \right)}^{-7}}={{\left( \frac{4}{9} \right)}^{2x-1}}\]

or  \[{{\left( \frac{4}{9} \right)}^{4+(-7)}}={{\left( \frac{4}{9} \right)}^{2x-1}}\]

or \[{{\left( \frac{4}{3} \right)}^{-3}}={{\left( \frac{4}{9} \right)}^{2x-1}}\]

By comparing with powers then,

\[-~~3=2x1\]

\[2x=-3+1\]

or \[~2x=2\]

or \[x=-\frac{2}{2}=-1\]

Hence,\[x=1\]