​$$\text{Find the value of } \left[ \left(\frac{3}{2}\right)^{-9} \times \left(\frac{2}{3}\right)^{-6} \times \left(\frac{1}{4}\right)^{-3} \right]^{-7}$$​​

Correct option is D

Given:

​$$\left[ \left(\frac{3}{2}\right)^{-9} \times \left(\frac{2}{3}\right)^{-6} \times \left(\frac{1}{4}\right)^{-3} \right]^{-7}$$

Formula Used:

a^-n = $$\left(\frac1a\right)^n$$ ; (a^m)ⁿ = a^mn

Solution:

​$$\left[ \left(\frac{3}{2}\right)^{-9} \times \left(\frac{3}{2}\right)^{6} \times 4^3\right]^{-7}$$

=$$\left[ \left(\frac{3}{2}\right)^{-9+6} \times 4^3\right]^{-7}$$

=$$\left[ \left(\frac{3}{2}\right)^{-3} \times \left(\frac14\right)^{-3}\right]^{-7}$$

=$$\left[ \left(\frac{3}{2} \times \frac14\right)^{-3}\right]^{-7}$$

=$$\left[ \left(\frac{3}8\right)^{-3}\right]^{-7}$$

$$= \left(\frac{3}{8}\right)^{-3\times -7} \\ \ \\= \left(\frac{3}{8}\right)^{21}$$​​