In technical fields like engineering and resource management, simplifying ratios of growth factors is a common task. Match each algebraic expression involving rational exponents with its correctly simplified equivalent according to the Quotient Property for Exponents.

A logistics coordinator uses the expression $\frac{x^{\frac{2}{3}}}{x^{\frac{5}{3}}}$ to compare the processing speeds of two automated systems. According to the Quotient Property for Exponents, which of the following is the most simplified form of this expression?

In a data analysis report, a researcher uses the expression $\frac{x^{\frac{1}{3}}}{x^{\frac{5}{3}}}$ to represent a specific growth ratio. True or False: According to the Quotient Property for Exponents, this expression simplifies to $\frac{1}{x^{\frac{4}{3}}}$.

A technical documentation specialist is simplifying the mathematical model $\frac{x^{\frac{1}{3}}}{x^{\frac{5}{3}}}$ for a system performance manual. Arrange the steps below in the correct order to simplify this expression using the Quotient Property for Exponents.

In a logistics optimization model, a technician simplifies the ratio of two performance factors represented by $\frac{x^{\frac{1}{3}}}{x^{\frac{5}{3}}}$. According to the Quotient Property for Exponents, this expression simplifies to $\frac{1}{x^k}$. The value of the exponent $k$ is ____.