$\begin{aligned} The value of \\ \frac{{(243)}^{0.13} \times {(243)}^{0.07}}{{(7)}^{0.25} \times {(49)}^{0.075} \times {(343)}^{0.2}} is = ? \end{aligned}$ $\eqalign{ & {\text{The value of}} \cr & \frac{{{{\left( {243} \right)}^{0.13}} \times {{\left( {243} \right)}^{0.07}}}}{{{{\left( 7 \right)}^{0.25}} \times {{\left( {49} \right)}^{0.075}} \times {{\left( {343} \right)}^{0.2}}}}{\text{is = ?}} \cr}$

1³/₇
2⁷/₃
3²/₇
4⁷/₂

Answer:- 1

Explanation:-

Solution:

\begin{aligned} \frac{{(243)}^{0.13} \times {(243)}^{0.07}}{{(7)}^{0.25} \times {(49)}^{0.075} \times {(343)}^{0.2}} \\ = \frac{{(243)}^{(0.13 + 0.07)}}{{(7)}^{0.25} \times {(7^{2})}^{0.075} \times {(7^{3})}^{0.2}} \\ = \frac{{(243)}^{0.2}}{{(7)}^{0.25} \times {(7)}^{(2 \times 0.075)} \times {(7)}^{(3 \times 0.2)}} \\ = \frac{{(3^{5})}^{0.02}}{{(7)}^{0.25} \times {(7)}^{0.15} \times {(7)}^{0.6}} \\ = \frac{{(3)}^{(5 \times 0.2)}}{{(7)}^{(0.25 + 0.15 + 0.6)}} \\ = \frac{3^{1}}{7^{1}} \\ = \frac{3}{7} \end{aligned}

$\eqalign{ & \frac{{{{\left( {243} \right)}^{0.13}} \times {{\left( {243} \right)}^{0.07}}}}{{{{\left( 7 \right)}^{0.25}} \times {{\left( {49} \right)}^{0.075}} \times {{\left( {343} \right)}^{0.2}}}} \cr & = \frac{{{{\left( {243} \right)}^{\left( {0.13 + 0.07} \right)}}}}{{{{\left( 7 \right)}^{0.25}} \times {{\left( {{7^2}} \right)}^{0.075}} \times {{\left( {{7^3}} \right)}^{0.2}}}} \cr & = \frac{{{{\left( {243} \right)}^{0.2}}}}{{{{\left( 7 \right)}^{0.25}} \times {{\left( 7 \right)}^{\left( {2 \times 0.075} \right)}} \times {{\left( 7 \right)}^{\left( {3 \times 0.2} \right)}}}} \cr & = \frac{{{{\left( {{3^5}} \right)}^{0.02}}}}{{{{\left( 7 \right)}^{0.25}} \times {{\left( 7 \right)}^{0.15}} \times {{\left( 7 \right)}^{0.6}}}} \cr & = \frac{{{{\left( 3 \right)}^{\left( {5 \times 0.2} \right)}}}}{{{{\left( 7 \right)}^{\left( {0.25 + 0.15 + 0.6} \right)}}}} \cr & = \frac{{{3^1}}}{{{7^1}}} \cr & = \frac{3}{7} \cr}$