Explanation:
x to the power of 1/2 means sqrt(x) (square root).
so, 9 to the power of 1/2 = sqrt(9) = 3.
therefore, we are looking for expressions that are also 3.
3 to the power of 1/2 = sqrt(3).
sqrt(3) × sqrt(3) = 3
this is correct
27 = 3×3×3
so, the cubic root would be 3 and correct, but the 6th root is incorrect.
27 to the power of 1/3 means the cubic root of 27, which is 3. but this is still multiplied by 3 to the power of 1/2 (sqrt(3)). so, the result is not 3 and therefore incorrect.
81 to the power of 1/2 = sqrt(81) = 9
9×9 = 81 and not 3. this is therefore incorrect.
729 to the power of 1/6 is the 6th root of 729, which is 3. this is therefore correct.
81 to the power of 1/3 means the cubic root of 81. but 81 is not a cubic number, and also multiplying this again with the cubic root of 81 is not 3. this is therefore incorrect.
81 to the power of 1/4 means the 4th root of 81. which is the square root of the square root.
the square root of 81 = 9. and the square root of 9 = 3.
3×3 = 9 and not 3. this is therefore incorrect.