The expression you provided is:

$\sqrt{5x} \cdot \sqrt{15x^5}$

To simplify this, we can first combine the square roots under one radical, as follows:

$\sqrt{5x} \cdot \sqrt{15x^5} = \sqrt{(5x) \cdot (15x^5)}$

Now, multiply the terms inside the square root:

$= \sqrt{75x^6}$

Next, we simplify the square root:

$\sqrt{75x^6} = \sqrt{25 \cdot 3 \cdot x^6} = \sqrt{25} \cdot \sqrt{3} \cdot \sqrt{x^6}$

We know that $\sqrt{25} = 5$ and $\sqrt{x^6} = x^3$, so the expression becomes:

$5x^3 \cdot \sqrt{3}$

Thus, the simplified form of the expression is:

$5x^3\sqrt{3}$

Would you like more details on any part of the solution?

Here are some related questions to deepen understanding:

1. How do you simplify square roots with variables?
2. What are the rules for multiplying square roots?
3. How do you simplify the square root of a product of numbers?
4. What is the process of rationalizing a square root?
5. How do you deal with fractional exponents?

Tip: Always break down square roots by separating perfect squares from non-perfect squares to simplify calculations.