How do you simplify root3(1/7)?

1 Answer
Fleur
Aug 15, 2017

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.

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