Question

By what number should `((-3)/2)^-3` be divided so that the quotient may be `(4/27)^-2`?

Answer 1

Let `((-3)/2)^-3` be divided by x to get `(4/27)^-2` as quotient.

Then, `((-3)/2)^-3 ÷ x = (4/27)^-2`

⇒ `x = ((-3)/2)^-3 ÷ (2^2/3^3)^-2 = ((-3)/2)^-3 ÷ (2)^-4/(3)^-6`

= `((-3)/2)^-3 xx (3)^-6/(2)^-4`

= `((-3)^-3 xx (3)^-6)/(2^-3 xx 2^-4`    ...[∵ a^m × aⁿ = a^{m + n}]

= `3^-9/2^-7` 

= `2^7/3^9`  ...`[∵ a^-m = 1/a^m  "and"  (a^m)^n = (a)^(mn)]`