The mathematical expression in the image is:

[ \sqrt[3]{7xy^2} \cdot \sqrt[3]{11x^{10}} ]

Step-by-step Solution:

1. Combine the cube roots: Using the property of exponents: [ \sqrt[3]{a} \cdot \sqrt[3]{b} = \sqrt[3]{a \cdot b}, ] we can combine the two cube roots: [ \sqrt[3]{7xy^2} \cdot \sqrt[3]{11x^{10}} = \sqrt[3]{(7xy^2)(11x^{10})}. ]

2. Multiply the terms inside the cube root: Multiply the constants and the variables: $(7)(11) = 77, \quad x \cdot x^{10} = x^{11}, \quad y^2 \text{ remains unchanged}.$ So the expression becomes: $\sqrt[3]{77x^{11}y^2}.$

3. Simplify the cube root: To simplify, look for terms that can be expressed as cubes:

   • $x^{11}$ can be split as $x^{9} \cdot x^2$, since $x^9 = (x^3)^3$, which is a perfect cube.
   • The rest ($x^2$, $y^2$, and 77) remain under the cube root.

   Therefore: [ \sqrt[3]{77x^{11}y^2} = x^3 \cdot \sqrt[3]{77x^2y^2}. ]

Final Answer:

$x^3 \cdot \sqrt[3]{77x^2y^2}.$

Let me know if you would like further explanation or have additional questions!
Here are five related questions to deepen your understanding:

1. How do cube roots differ from square roots when simplifying expressions?
2. Why can terms like $x^{9}$ be factored out from cube roots?
3. Can $y^2$ or $x^2$ be further simplified in this problem? Why or why not?
4. What are other examples of expressions involving cube roots that can be simplified similarly?
5. How would this change if it were a fourth root instead of a cube root?

Tip: Always look for perfect powers (like $x^9$) when simplifying roots to make expressions more compact!