Question

সরল কর: $\frac{3^{m+1}}{\left(3^{m}\right)^{m-1}}\div\frac{9^{m+1}}{\left(3^{m-1}\right)^{m+1}}$

Answer 1

$\frac{3^{m+1}}{\left(3^{m}\right)^{m-1}}\div\frac{9^{m+1}}{\left(3^{m-1}\right)^{m+1}}$

$=\frac{3^{m+1}}{\left(3^{m}\right)^{m-1}}\div\frac{\left(3^2\right)^{m+1}}{\left(3^{m-1}\right)^{m+1}}$

$=\frac{3^{m+1}}{3^{m\left(m-1\right)}}\div\frac{3^{2\left(m+1\right)}}{3^{\left(m-1\right)\left(m+1\right)}}$

$=\frac{3^{m+1}}{3^{m^2-m}}\div\frac{3^{2m+2}}{3^{m^2-1^2}}$

$=3^{\left(m+1\right)-(m^2-m)}\div3^{\left(2m+2\right)-(m^2-1)}$

$=3^{m+1-m^2+m}\div3^{2m+2-m^2+1}$

$=3^{2m+1-m^2}\div3^{2m+3-m^2}$

$=3^{\left(2m+1-m^2\right)-\left(2m+3-m^2\right)}$

$=3^{2m+1-m^2-2m-3+m^2}$

$=3^{-2}$

$=\frac{1}{3^2}$

$=\frac19$ [Answer]