https://file.aiquickdraw.com/mathbot/file/17270444463167fw9es9v.jpg

Let's simplify the given expression, which is:

\[
\sqrt{10x^3} \cdot \sqrt{5x^2}
\]

We can first combine the square roots using the property:

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

So we get:

\[
\sqrt{10x^3 \cdot 5x^2} = \sqrt{50x^5}
\]

Next, simplify the square root:

\[
\sqrt{50x^5} = \sqrt{25 \cdot 2 \cdot x^4 \cdot x} = \sqrt{25x^4} \cdot \sqrt{2x}
\]

Since \(\sqrt{25x^4} = 5x^2\), the expression becomes:

\[
5x^2 \cdot \sqrt{2x}
\]

Thus, the simplified form is:

\[
5x^2 \sqrt{2x}
\]

This matches the last answer choice.

Would you like more details or have any questions?

### Related Questions:
1. How do you simplify products of square roots in general?
2. What are the properties of exponents and how are they used in simplifications?
3. What is the general rule for taking square roots of variables with exponents?
4. How do you factor square roots to simplify expressions?
5. How would the process change if there were cube roots instead of square roots?

### Tip:
When dealing with square roots, always try to simplify the radicand (the number inside the square root) by factoring out perfect squares.

Simplify the given expression: √(10x³) * √(5x²)

This math problem explains how to simplify the expression √(10x³) * √(5x²) by combining square roots and simplifying exponents. Suitable for high school students (Grades 9-11), covering key concepts such as multiplication of square roots and exponent properties.

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