To find the equivalent expression for [tex]\(\sqrt{10 x^2} \cdot \sqrt{5 x}\)[/tex] when [tex]\(x \geq 0\)[/tex], we'll follow a step-by-step approach:
1. Simplify Each Square Root:
- [tex]\(\sqrt{10 x^2}\)[/tex]
- [tex]\(\sqrt{5 x}\)[/tex]
Let's handle them one at a time.
[tex]\[
\sqrt{10 x^2} = \sqrt{10} \cdot \sqrt{x^2}
\][/tex]
Since [tex]\(\sqrt{x^2} = x\)[/tex] (because [tex]\(x \geq 0\)[/tex]), we have:
[tex]\[
\sqrt{10 x^2} = \sqrt{10} \cdot x
\][/tex]
Next, consider:
[tex]\[
\sqrt{5 x} = \sqrt{5} \cdot \sqrt{x}
\][/tex]
2. Multiply the Simplified Expressions:
We now multiply the two simplified expressions:
[tex]\[
\sqrt{10 x^2} \cdot \sqrt{5 x} = (\sqrt{10} \cdot x) \cdot (\sqrt{5} \cdot \sqrt{x})
\][/tex]
This can be rewritten as:
[tex]\[
\sqrt{10} \cdot x \cdot \sqrt{5} \cdot \sqrt{x}
\][/tex]
We can combine the square root terms:
[tex]\[
\sqrt{10} \cdot \sqrt{5} \cdot x \cdot \sqrt{x} = \sqrt{10 \cdot 5} \cdot x \cdot \sqrt{x}
\][/tex]
Simplify inside the square root:
[tex]\[
\sqrt{50} \cdot x \cdot \sqrt{x}
\][/tex]
Notice that [tex]\(\sqrt{50}\)[/tex] can be further simplified:
[tex]\[
\sqrt{50} = \sqrt{25 \cdot 2} = \sqrt{25} \cdot \sqrt{2} = 5 \cdot \sqrt{2}
\][/tex]
Substituting this back in, we get:
[tex]\[
5 \cdot \sqrt{2} \cdot x \cdot \sqrt{x}
\][/tex]
3. Combine [tex]\(x\)[/tex] Terms:
Combine the [tex]\(x\)[/tex] terms:
[tex]\[
5 \cdot \sqrt{2} \cdot x \cdot \sqrt{x} = 5 x \cdot \sqrt{2} \cdot \sqrt{x} = 5 x \cdot \sqrt{2 x}
\][/tex]
Thus,
[tex]\[
\sqrt{10 x^2} \cdot \sqrt{5 x} = 5 x \cdot \sqrt{2 x}
\][/tex]
This matches choice D:
[tex]\[
\boxed{5 x \sqrt{2 x}}
\][/tex]
