The answer is 1.) 6^1/6
The third root of something is the same as an exponent to the 1/3 power.
(When an exponent is a fraction, the numerator is powers and the denominator finds the root. For example: 3 to the power of 4/2 is multiplied by itself 4 times (81) then the denominator, 2, means we find the square root of it (9). 4/2=2, and 3 to the 2nd power is also 9.)
So that would be 6 to the 1/3 power, then you find the square root of that, so it's (6 to the 1/3 power) to the 1/2 power, and according to the Power Rule (a power on top of a power means you multiply the two powers), that would end up being 1/6.
![\sqrt{ \sqrt[3]{6} } = \sqrt{ {6}^{ \frac{1}{3} } } = (6 { \frac{1}{3} })^{ \frac{1}{2} } = {6}^{ \frac{1}{3} \times \frac{1}{2} } = {6}^{ \frac{1}{6} }](https://tex.z-dn.net/?f=%5Csqrt%7B%20%20%5Csqrt%5B3%5D%7B6%7D%20%20%7D%20%20%3D%20%20%5Csqrt%7B%20%7B6%7D%5E%7B%20%5Cfrac%7B1%7D%7B3%7D%20%7D%20%7D%20%20%3D%20%286%20%7B%20%5Cfrac%7B1%7D%7B3%7D%20%7D%29%5E%7B%20%5Cfrac%7B1%7D%7B2%7D%20%7D%20%20%3D%20%20%7B6%7D%5E%7B%20%5Cfrac%7B1%7D%7B3%7D%20%20%5Ctimes%20%20%5Cfrac%7B1%7D%7B2%7D%20%7D%20%20%3D%20%20%7B6%7D%5E%7B%20%5Cfrac%7B1%7D%7B6%7D%20%7D%20)
The third root of something is the same as an exponent to the 1/3 power.
(When an exponent is a fraction, the numerator is powers and the denominator finds the root. For example: 3 to the power of 4/2 is multiplied by itself 4 times (81) then the denominator, 2, means we find the square root of it (9). 4/2=2, and 3 to the 2nd power is also 9.)
So that would be 6 to the 1/3 power, then you find the square root of that, so it's (6 to the 1/3 power) to the 1/2 power, and according to the Power Rule (a power on top of a power means you multiply the two powers), that would end up being 1/6.