Let's simplify the given expression:

[ \frac{x^2 \left( \sqrt[3]{x \cdot x^2 + x^4} \right)}{\sqrt[4]{x^5}} ]

To break it down, we can simplify each part step-by-step.

Step 1: Simplify the Expression Inside the Cube Root

Inside the cube root, we have: $x \cdot x^2 + x^4$ This simplifies as follows: $x \cdot x^2 = x^3$ So we get: $x^3 + x^4 = x^3(1 + x)$ Thus, the cube root part becomes: $\sqrt[3]{x^3 (1 + x)} = x (1 + x)^{1/3}$

Step 2: Substitute Back into the Main Expression

Now, the expression becomes: $\frac{x^2 \cdot x (1 + x)^{1/3}}{\sqrt[4]{x^5}}$

Simplify the numerator: $x^2 \cdot x = x^3$ So we have: $\frac{x^3 (1 + x)^{1/3}}{\sqrt[4]{x^5}}$

Step 3: Simplify the Denominator

The denominator $\sqrt[4]{x^5}$ can be rewritten as: $x^{5/4}$

Step 4: Divide the Terms

Now we have: $\frac{x^3 (1 + x)^{1/3}}{x^{5/4}}$

Subtract the exponents of $x$: $x^{3 - 5/4} (1 + x)^{1/3} = x^{12/4 - 5/4} (1 + x)^{1/3} = x^{7/4} (1 + x)^{1/3}$

Final Answer

The simplified expression is: $x^{7/4} (1 + x)^{1/3}$

Would you like further details or have any questions about the steps?

Here are five related questions to expand on this topic:

1. How do we simplify expressions involving roots and exponents in general?
2. What are the properties of exponents and roots that make simplification possible?
3. How does the fractional exponent notation relate to radical expressions?
4. What are common mistakes to avoid when simplifying expressions like this?
5. How can expressions with mixed roots be handled differently?

Tip: When simplifying expressions with roots, try converting roots to fractional exponents, as it often makes the simplification process more straightforward.