Question

Calculate:

`8^(2/3)/(root(3)(2 xx 4^-5)`

Answer 1

Given,

`8^(2/3)/(root(3)(2 xx 4^-5)`

We need to simplify the given terms.

Thus, `8^(2/3)/(root(3)(2 xx 4^-5)`

⇒ `(2^3)^(2/3)/((2 xx 4^(-5 xx 1/3))`  ...`[∴ root(n)(a) = a^(1/n)]` 

⇒ `(2)^(3 xx 2/3)/((2^(1/3) xx 4^(-5 xx 1/3))`  ...[∴ (aⁿ)^m = a^nm, aⁿ × bⁿ = (ab)ⁿ]

⇒ `(2)^2/((2^(1/3) xx 4^(-5/3))`

⇒ `(2)^2/(2^(1/3) xx (2^2)^(- 5/3)`

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

⇒ `(2)^2/(2^(1/3) xx (2)^((-10)/3)`  ...[∴ (aⁿ)^m = a^nm]

⇒ `(2)^2/2^((1/3 - 10/3))`  ...[∴ aⁿ × a^m = a^{n + m}]

⇒ `(2)^2/2^((-9)/3) = (2)^2/2^-3`

⇒ (2)² × 2³  ...`[∴ a^n = 1/a^-n]`

⇒ 2⁵ = 32   ...[∴ aⁿ × a^m = a^{n + m}]

Hence, the required is 32.