Question

Simplify and express with a positive index.

`root(5)(32a^-10) xx 1/a^-3`

Answer 1

Given,

`root(5)(32a^-10) xx 1/a^-3`

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

Thus, `root(5)(32a^-10) xx 1/a^-3`

⇒ `(32a^-10)^(1/5) xx 1/a^-3`  ...`[∴ root(n)(a) = a^(1/n)]`

⇒ `(2^5)^(1/5) xx (a^-10)^(1/5) xx 1/a^-3`  ...[∴ (ab)ⁿ = aⁿ × bⁿ]

⇒ `(2)^(5 xx 1/5) xx (a)^(-10 xx 1/5) xx 1/a^-3`  ...[∴ (aⁿ)^m = a^nm]

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

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

⇒ `(2) xx (a)^1 = 2a`

Hence, `root(5)(32a^-10) xx 1/a^-3 = 2a`