Question

Simplify:

`(27/125)^(1/3) xx [(3/2)^-2 ÷ (2/5)^3]`

Answer 1

Given,

`(27/125)^(1/3) xx [(3/2)^-2 ÷ (2/5)^3]`

We have to simplify the given expression.

Thus, `(27/125)^(1/3) xx [(3/2)^-2 ÷ (2/5)^3]`

⇒ `(3^3/5^3)^(1/3) xx [(3/2)^-2 ÷ (2/5)^3]`

⇒ `(3/5)^(3 xx 1/3) xx [(2/3)^2 ÷ (2/5)^3]`  ...`[∴ (x/y)^-n = (y/x)^n]`

⇒ `3/5 xx (2^2/3^2) ÷ (2^3/5^3)`  ...`[∴ (x/y)^n = x^n/y^n]`

⇒ `3/5 xx (2^2/3^2) ÷ (5^3/2^3)`  ...`[∴ (x/y)^n ÷ (p/q)^m = (x/y)^n xx (q/p)^m]` 

⇒ `3 xx 3^-2 xx 2^2 xx 2^-3 xx 5^3 xx 5^-1`  ...[∴ xⁿ × x^m = x^{n + m}]

⇒ `3^(1 - 2) xx 2^(2 - 3) xx 5^(3 - 1)`

⇒ `3^-1 xx 2^-1 xx 5^2`

⇒ `5^2/(3 xx 2) = 25/6 = 4 1/6`  ...`[∴ x^-n = 1/x^n]`

Hence, `(27/125)^(1/3) xx [(3/2)^-2 ÷ (2/5)^3] = 4 1/6`.