Question

How do you simplify `root3(1/7)`?

Answer 1

`root 3 49/ 7`

Explanation:

Since `x^(m/n) = rootn (x^m)`, we can write `root(3) (1/7)` as `(1/7)^(1/3)`.

According to the power of a quotient rule, `(a/b)^m = a^m/b^m`. Thus, we can say `(1/7)^(1/3)= 1^(1/3) / 7 ^ (1/3)`.

From here, we can say `1^(1/3) / 7 ^ (1/3) = 1/ 7^(1/3)`, since `1` raised to any power is `1`.

We are left with `1/ 7^(1/3)` or `1/root3 7`.

However, we cannot have a radical in the denominator. To rationalize this expression, we must try to make the denominator `7`. To do this, multiply both the numerator and denominator by `7^(2/3) / 7^(2/3)` or `root3 (7^2)/ root3 (7^2)`, respectively.

`1/ 7^(1/3) * color(blue)(7^(2/3)/7^(2/3)) = 7^(2/3) / 7^(3/3) = 7^(2/3) /7 = root3 (7^2) /7 = root3 49 /7`

`1/ root 3 7 * color(blue)(root3 (7^2)/ root3 (7^2)) = root3 49 / root 3 343 = root 3 49/ 7`

So, `(1/7)^(1/3)` simplifies to `root 3 49/ 7`.