Question

Simplify and express with a positive index.

`root(5)(32a^5b^10c^-15)`

Answer 1

Given,

`root(5)(32a^5b^10c^-15)`

We need to simplify and express with a positive index the given terms.

We know that if a, b, c are any non-zero integers and m and n are the whole numbers, then we have,

`root(5)(32a^5b^10c^-15)`

⇒ `(32a^5b^10c^-15)^(1/5)`   ...`[∴ root(n)(a) = a^(1/n)]`

⇒ `32^(1/5) xx (a^5)^(1/5) xx (b^10)^(1/5) xx (c^-15)^(1/5)`  ...[∴ (ab)ⁿ = aⁿ × bⁿ]

⇒ `(2^5)^(1/5) xx (a^5)^(1/5) xx (b^10)^(1/5) xx (c^-15)^(1/5)`

⇒ `(2)^(5 xx 1/5) xx (a)^(5 xx 1/5) xx (b)^(10 xx 1/5) xx (c)^(-15 xx 1/5)`  ...[∴ (aⁿ)^m = a^nm]

⇒ `(2) xx (a) xx (b)^2 xx (c)^-3`

⇒ `2ab^2 xx 1/c^3`  ...`[∴ a^-n = 1/a^n]`

⇒ `(2ab^2)/c^3`

Hence, `root(5)(32a^5b^10c^-15) = (2ab^2)/c^3`.