Question

Simplify and express with a positive index.

`root(4)(81a^12b^-16)`

Answer 1

Given,

`root(4)(81a^12b^-16)`

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

Thus, `root(4)(81a^12b^-16)`

= `root(4)(3^4a^12b^-16)`

= `(3^4a^12b^-16)^(1/4)`   ...`[∴ root(n)(a) = a^(1/n)]`

The power `1/4` is taken for each factor inside the bracket.

= `(3^4)^(1/4) xx (a^12)^(1/4) xx (b^-16)^(1/4)`

= `3^(4 xx 1/4) xx a^(12 xx 1/4) xx b^(-16 xx 1/4)`  ...[(a^m)ⁿ = a^mn]

= `3^1 xx a^3 xx b^-4`

= `(3a^3)/b^4`  ...`[∴ a^-n = 1/a^n]`

Hence, `root(4)(81a^12b^-16) = (3a^3)/b^4`.