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

1 Answer
Aug 15, 2017

Answer:

#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#.