Let's break down the given equation step by step:

The equation is:

$\left( x^2 \right)^{\frac{1}{5}} \cdot \sqrt[5]{x} = x^a$

Step 1: Simplify $\left( x^2 \right)^{\frac{1}{5}}$

Using the property of exponents $(x^m)^n = x^{m \cdot n}$, we can simplify $\left( x^2 \right)^{\frac{1}{5}}$ as follows:

$\left( x^2 \right)^{\frac{1}{5}} = x^{2 \cdot \frac{1}{5}} = x^{\frac{2}{5}}$

Step 2: Simplify $\sqrt[5]{x}$

The fifth root of $x$ is the same as raising $x$ to the power of $\frac{1}{5}$:

$\sqrt[5]{x} = x^{\frac{1}{5}}$

Step 3: Combine both terms

Now, we can combine the terms on the left side of the equation:

$x^{\frac{2}{5}} \cdot x^{\frac{1}{5}}$

Using the property of exponents $x^m \cdot x^n = x^{m+n}$, we add the exponents:

$x^{\frac{2}{5} + \frac{1}{5}} = x^{\frac{3}{5}}$

Step 4: Set the equation equal to $x^a$

Now the equation looks like this:

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

Step 5: Solve for $a$

Since the bases are the same (both $x$), we can set the exponents equal to each other:

$\frac{3}{5} = a$

Thus, the value of $a$ is:

$a = \frac{3}{5}$

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Would you like further clarification on any of these steps? Here are a few related questions to explore:

1. How does the exponent rule $(x^m)^n = x^{m \cdot n}$ work in more complex examples?
2. Can you apply the same process to roots with other powers, such as square roots or cube roots?
3. What happens if the exponents have negative values in equations like this?
4. How can we use these exponent properties when solving equations with fractional exponents?
5. How do you solve equations that involve both multiplication and division of exponents?

Tip: When working with exponents, always remember the basic rules, especially for powers and roots. Simplifying terms step by step helps avoid mistakes!