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We need to simplify the left-hand side of the equation to find the exponent $a$ in terms of $x^a$.

Given expression:

$\frac{x^{\frac{5}{3}}}{x^{\frac{6}{5}}} = x^a$

Step 1: Apply the Quotient Rule of Exponents

The quotient rule states: $\frac{x^m}{x^n} = x^{m-n}$ Applying this rule: $x^{\frac{5}{3} - \frac{6}{5}} = x^a$

Step 2: Find a Common Denominator

The denominators are 3 and 5. The least common denominator is 15. Convert each fraction:

$\frac{5}{3} = \frac{25}{15}, \quad \frac{6}{5} = \frac{18}{15}$

Now, subtract: $\frac{25}{15} - \frac{18}{15} = \frac{7}{15}$

Step 3: Conclusion

$x^{\frac{7}{15}} = x^a$

Thus, the value of $a$ is: $\frac{7}{15}$

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Do you need more clarification or a step-by-step breakdown? Here are some related questions:

1. How does the quotient rule for exponents work in general?
2. Can you explain how to find a common denominator in fractions?
3. What happens if the bases are different in exponent rules?
4. How would this problem change if there were coefficients in front of $x$?
5. How do negative exponents affect the calculation?

Tip: Always express exponents in a common denominator before performing addition or subtraction.