Question

Simplify:

`(((-2)/3)^-2)^3 xx (1/3)^-4 xx 3^-1 xx 1/6`

Answer 1

Using laws of exponents,

(a^m)ⁿ = (a)^{m × n}, `a^-m = 1/a^m`, a^m × aⁿ = a^{m + n} and a^m ÷ aⁿ = a^{m – n}  ...[∵ a is non-zero integer]

∴ `(((-2)/3)^-2)^3 xx (1/3)^-4 xx 3^-1 xx 1/6 = ((-2)/3)^((-2) xx 3) xx (3)^4 xx 1/3 xx 1/6`

= `((-2)/3)^-6 xx 3^4 xx 1/3 xx 1/(2 xx 3)`  ...[∵ 6 = 2 × 3]

= `(3/(-2))^6 xx 3^4 xx 1/(3 xx 2 xx 3) `

= `(3)^6/(-2)^6 xx 3^4 xx 1/(2^1 xx 3^2)`

= `(3)^(6 + 4)/((2)^(6 + 1) xx 3^2)`  ...[(–a^m) = a^m, If m is an even number]

= `(3)^10/(2^7 xx 3^2)`

= `3^(10 - 2)/2^7`

= `3^8/2^7`