Simplify the expression. \[ y^{-\frac{4}{3}} y^{\frac{1}{5}} y^{\frac{1}{6}} \] Write your answer using only positive exponents. Assume that all variables are positive real numbers.

To simplify the expression \( y^{-\frac{4}{3}} y^{\frac{1}{5}} y^{\frac{1}{6}} \), we can combine the exponents since the bases (which are all \( y \)) are the same. We do this by adding the exponents: \[ -\frac{4}{3} + \frac{1}{5} + \frac{1}{6} \] First, we need a common denominator. The least common multiple of 3, 5, and 6 is 30. We will convert each fraction to have a denominator of 30: \[ -\frac{4}{3} = -\frac{4 \times 10}{3 \times 10} = -\frac{40}{30} \] \[ \frac{1}{5} = \frac{1 \times 6}{5 \times 6} = \frac{6}{30} \] \[ \frac{1}{6} = \frac{1 \times 5}{6 \times 5} = \frac{5}{30} \] Now we can add these fractions together: \[ -\frac{40}{30} + \frac{6}{30} + \frac{5}{30} = \frac{-40 + 6 + 5}{30} = \frac{-40 + 11}{30} = \frac{-29}{30} \] This means that: \[ y^{-\frac{4}{3}} y^{\frac{1}{5}} y^{\frac{1}{6}} = y^{-\frac{29}{30}} \] Since we want to express this with only positive exponents, we can rewrite it as: \[ \frac{1}{y^{\frac{29}{30}}} \] So, the simplified expression is: \[ \frac{1}{y^{\frac{29}{30}}} \]