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rewrite with positive exponents that cannot be simplified 

To rewrite the given expression with positive exponents, we'll follow these steps:

1. **Identify the expression:**

   \[
   \frac{x^2 y^{-3} z^{\frac{3}{5}}}{x^{-5} y^{\frac{5}{8}} z^{\frac{1}{3}}}
   \]

2. **Simplify the expression:**

   - For the \(x\) terms: \(x^2 \div x^{-5} = x^{2 - (-5)} = x^{2 + 5} = x^7\)
   - For the \(y\) terms: \(y^{-3} \div y^{\frac{5}{8}} = y^{-3 - \frac{5}{8}} = y^{-\frac{24}{8} - \frac{5}{8}} = y^{-\frac{29}{8}}\)
   - For the \(z\) terms: \(z^{\frac{3}{5}} \div z^{\frac{1}{3}} = z^{\frac{3}{5} - \frac{1}{3}}\)

     To subtract these fractions, find a common denominator:

     \[
     \frac{3}{5} = \frac{9}{15}, \quad \frac{1}{3} = \frac{5}{15}
     \]

     So,

     \[
     \frac{9}{15} - \frac{5}{15} = \frac{4}{15}
     \]

     Therefore, \(z^{\frac{3}{5}} \div z^{\frac{1}{3}} = z^{\frac{4}{15}}\)

3. **Combine the results:**

   \[
   \frac{x^2 y^{-3} z^{\frac{3}{5}}}{x^{-5} y^{\frac{5}{8}} z^{\frac{1}{3}}} = x^7 y^{-\frac{29}{8}} z^{\frac{4}{15}}
   \]

4. **Rewrite with positive exponents:**

   To ensure all exponents are positive, move \(y^{-\frac{29}{8}}\) to the denominator:

   \[
   x^7 z^{\frac{4}{15}} \cdot \frac{1}{y^{\frac{29}{8}}} = \frac{x^7 z^{\frac{4}{15}}}{y^{\frac{29}{8}}}
   \]

   So, the expression with positive exponents is:

   \[
   \frac{x^7 z^{\frac{4}{15}}}{y^{\frac{29}{8}}}
   \]

Would you like further details on the steps taken? Here are some related questions you might find helpful:

1. How do you add or subtract fractions with different denominators?
2. How do you convert a negative exponent into a positive exponent?
3. What are the rules for simplifying expressions involving exponents?
4. How do you simplify complex fractions?
5. How can exponent properties be used to solve algebraic equations?

**Tip:** Always remember to find a common denominator when subtracting fractions in exponents, and convert negative exponents to positive by moving the base to the other side of the fraction (numerator to denominator or vice versa).

Learn how to rewrite expressions with positive exponents with a step-by-step solution. This problem involves exponents and fractional exponents, demonstrating how to simplify and convert negative exponents to positive. Suitable for high school students.

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