Final answer:
To simplify the expression 3x⁻¹/² y+2/2/3y-x⁻ ⁵/⁶, we can rewrite the expressions using the rules of exponents and then rationalize the denominators. The simplified expression is 3 sqrt(x) * (y+2) * (3y-x⁻ ⁵/⁶)/2.
Step-by-step explanation:
To simplify the expression 3x⁻¹/² y+2/2/3y-x⁻ ⁵/⁶, we can start by using the rules of exponents. We have 3x⁻¹/², which can be rewritten as 3/(x⁻¹/²) = 3/(1/sqrt(x)). Similarly, we have y+2/2/3y-x⁻ ⁵/⁶, which can be simplified as (y+2)/(2/3y-x⁻ ⁵/⁶).
Now, we can simplify the expression further by rationalizing the denominator of both parts. For the first part, we multiply the numerator and denominator by sqrt(x) to get 3 * sqrt(x) / 1 = 3 sqrt(x). For the second part, we multiply the numerator and denominator by the reciprocal of the denominator, which is (2/3y-x⁻ ⁵/⁶)⁻¹. This gives us (y+2) * (3y-x⁻ ⁵/⁶)/2.
Putting it all together, the simplified expression is 3 sqrt(x) * (y+2) * (3y-x⁻ ⁵/⁶)/2.