Question

Simplify and express with a positive index.

`(root(3)(343a^6b^-9))/(root(4)(16b^4))`

Answer 1

Given,

`(root(3)(343a^6b^-9))/(root(4)(16b^4))`

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

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

`(root(3)(343a^6b^-9))/(root(4)(16b^4))`

⇒ `(343a^6b^-9)^(1/3)/(16b^4)^(1/4)`   ...`[∴ root(n)(a) = a^(1/n)]`

⇒ `[((7^3)^(1/3) xx (a^6)^(1/3) xx (b^-9)^(1/3))/((4^2)^(1/4) xx (b^4)^(1/4))]`   ...[∴ (ab)ⁿ = aⁿ × bⁿ]

⇒ `[((7)^(3 xx 1/3) xx (a)^(6 xx 1/3) xx (b)^(-9 xx 1/3))/((4)^(2 xx 1/4) xx (b)^(4 xx 1/4))]`   ...[∴ (aⁿ)^m = a^nm]

⇒ `[(7 xx a^2 xx b^-3)/((4)^(1/2) xx b)]`

⇒ `[(7a^2b^-3)/((2^2)^(1/2) xx b)]`

⇒ `(7a^2b^-3)/(2b)`  ...`[∴ a^-n = 1/a^n]`

⇒ `(7a^2)/(2b xx b^3)`

⇒ `(7a^2)/(2b^4)`

Hence, `(root(3)(343a^6b^-9))/(root(4)(16b^4)) = (7a^2)/(2b^4)`.