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

To simplify the expression, we will first combine the exponents of \( z \). We have: \[ z^{-\frac{4}{3}} z^{\frac{1}{6}} z^{\frac{1}{5}} = z^{-\frac{4}{3} + \frac{1}{6} + \frac{1}{5}} \] Now, we need to combine the exponents. The first step is to find a common denominator for the fractions \(-\frac{4}{3}\), \(\frac{1}{6}\), and \(\frac{1}{5}\). The least common multiple of the denominators \(3\), \(6\), and \(5\) is \(30\). Now we will convert each fraction: 1. \(-\frac{4}{3} = -\frac{4 \times 10}{3 \times 10} = -\frac{40}{30}\) 2. \(\frac{1}{6} = \frac{1 \times 5}{6 \times 5} = \frac{5}{30}\) 3. \(\frac{1}{5} = \frac{1 \times 6}{5 \times 6} = \frac{6}{30}\) Now we can sum them up: \[ -\frac{40}{30} + \frac{5}{30} + \frac{6}{30} = \frac{-40 + 5 + 6}{30} = \frac{-40 + 11}{30} = \frac{-29}{30} \] Thus, we have: \[ z^{-\frac{4}{3}} z^{\frac{1}{6}} z^{\frac{1}{5}} = z^{-\frac{29}{30}} \] To express the infinite fraction with positive exponents, we rewrite it as: \[ \frac{1}{z^{\frac{29}{30}}} \] Hence, the simplified expression with positive exponents is: \[ \frac{1}{z^{\frac{29}{30}}} \]