The equation shown is:

$\frac{\left(x^{\frac{1}{3}}\right)^5}{\sqrt[4]{x^5}} = x^a$

Step-by-step solution:

1. Simplify the numerator: $\left(x^{\frac{1}{3}}\right)^5 = x^{\frac{5}{3}}$ Using the power of a power rule: $(x^m)^n = x^{m \cdot n}$.

2. Simplify the denominator: $\sqrt[4]{x^5} = x^{\frac{5}{4}}$ Using the rule: $\sqrt[n]{x^m} = x^{\frac{m}{n}}$.

3. Write the entire expression: $\frac{x^{\frac{5}{3}}}{x^{\frac{5}{4}}}$ When dividing powers with the same base, subtract the exponents: $\frac{x^m}{x^n} = x^{m - n}$.

4. Subtract the exponents: $\frac{5}{3} - \frac{5}{4} = \frac{20}{12} - \frac{15}{12} = \frac{5}{12}$ So the expression simplifies to: $x^{\frac{5}{12}}$

5. Equate this to $x^a$: $x^{\frac{5}{12}} = x^a$

6. Conclusion: Since the bases are the same, the exponents must be equal: $a = \frac{5}{12}$

Thus, $a = \frac{5}{12}$.

Would you like more details or have any questions about this?

Here are some related questions for practice:

1. How do you apply the exponent rules in multiplication and division of powers?
2. Can you simplify $\sqrt[3]{x^6}$ using fractional exponents?
3. What happens if the base of the powers in the equation are different?
4. How would the solution change if the numerator had a different exponent?
5. What is the general form for converting radical expressions to exponents?

Tip: When solving equations with exponents, always use exponent rules like multiplication, division, and powers of powers to simplify expressions.