To solve this problem, we are tasked with finding the simplified product of:

[ \sqrt[3]{15} \cdot \sqrt[3]{9} ]

Step-by-step solution:

1. Understanding the expression: Both terms are cube roots, so we can combine them using the property of radicals:

   [ \sqrt[3]{a} \cdot \sqrt[3]{b} = \sqrt[3]{a \cdot b} ]

2. Applying this property:

   [ \sqrt[3]{15} \cdot \sqrt[3]{9} = \sqrt[3]{15 \cdot 9} ]

3. Multiplying inside the cube root:

   $15 \times 9 = 135$

   Thus, the expression becomes:

   $\sqrt[3]{135}$

4. Simplifying the cube root: We can try to factor 135 to see if any factors are perfect cubes.

   $135 = 3^3 \times 5$

   Therefore:

   [ \sqrt[3]{135} = 3 \sqrt[3]{5} ]

Thus, the simplified product is:

$3 \sqrt[3]{5}$

The correct answer is: $3 \sqrt[3]{5}$.

Would you like more details on any of the steps?

Here are 5 related questions you might find helpful:

1. How do you simplify expressions involving cube roots in general?
2. What is the difference between square roots and cube roots when simplifying products?
3. How can you determine if a number inside a radical can be simplified?
4. What are the properties of radicals when multiplying and dividing them?
5. How do you simplify higher-order radicals (e.g., fourth or fifth roots)?

Tip: When simplifying products involving radicals, always check if the numbers can be factored into powers that match the index of the root, as this helps with simplification.