Let's break down the simplification steps for each given expression in detail.
### Expression 1: [tex]\(\frac{\sqrt{5 x^7}}{\sqrt{3 x}} \cdot \sqrt{15 x^2}\)[/tex]
1. Simplify [tex]\(\frac{\sqrt{5 x^7}}{\sqrt{3 x}}\)[/tex]:
- [tex]\(\sqrt{5 x^7}\)[/tex] can be written as [tex]\(\sqrt{5} \cdot \sqrt{x^7}\)[/tex].
- [tex]\(\sqrt{3 x}\)[/tex] can be written as [tex]\(\sqrt{3} \cdot \sqrt{x}\)[/tex].
- Dividing these, we get [tex]\(\frac{\sqrt{5} \cdot \sqrt{x^7}}{\sqrt{3} \cdot \sqrt{x}} = \sqrt{\frac{5}{3}} \cdot \sqrt{\frac{x^7}{x}} = \sqrt{\frac{5}{3}} \cdot \sqrt{x^6} = \sqrt{\frac{5}{3}} \cdot x^3\)[/tex].
2. Now, multiply by [tex]\(\sqrt{15 x^2}\)[/tex]:
- [tex]\(\sqrt{15 x^2}\)[/tex] can be written as [tex]\(\sqrt{15} \cdot \sqrt{x^2} = \sqrt{15} \cdot x\)[/tex].
- Multiplying, we get: [tex]\(\sqrt{\frac{5}{3}} \cdot x^3 \cdot \sqrt{15} \cdot x\)[/tex].
3. Combine the radicals:
- [tex]\(\sqrt{\frac{5}{3}} \cdot \sqrt{15} = \sqrt{\frac{5}{3} \cdot 15} = \sqrt{25} = 5\)[/tex].
- Therefore, [tex]\(5 \cdot x^3 \cdot x = 5 \cdot x^4\)[/tex].
So, the simplified form of [tex]\(\frac{\sqrt{5 x^7}}{\sqrt{3 x}} \cdot \sqrt{15 x^2}\)[/tex] is [tex]\(\boxed{5 x^4}\)[/tex].
### Expression 2: [tex]\(\sqrt{25 x^8}\)[/tex]
- [tex]\(\sqrt{25 x^8}\)[/tex] can be written as [tex]\(\sqrt{25} \cdot \sqrt{x^8}\)[/tex].
- [tex]\(\sqrt{25} = 5\)[/tex], and [tex]\(\sqrt{x^8} = x^4\)[/tex].
So, [tex]\(\sqrt{25 x^8} = 5 x^4\)[/tex].
Thus, the simplified form of [tex]\(\sqrt{25 x^8}\)[/tex] is [tex]\(\boxed{5 x^4}\)[/tex].
### Expression 3: [tex]\(5 x^4\)[/tex]
This expression is already in its simplest form.
So, the simplified form of [tex]\(5 x^4\)[/tex] is [tex]\(\boxed{5 x^4}\)[/tex].
### Expression 4: [tex]\(5 x^2 \sqrt{x^6}\)[/tex]
- [tex]\(\sqrt{x^6} = x^3\)[/tex] because the square root of [tex]\(x^6\)[/tex] is [tex]\(x^3\)[/tex].
Therefore, [tex]\(5 x^2 \sqrt{x^6} = 5 x^2 \cdot x^3 = 5 x^5\)[/tex].
So, the simplified form of [tex]\(5 x^2 \sqrt{x^6}\)[/tex] is [tex]\(\boxed{5 x^5}\)[/tex].
### Expression 5: [tex]\(25 x^8\)[/tex]
This expression is already in its simplest form.
So, the simplified form of [tex]\(25 x^8\)[/tex] is [tex]\(\boxed{25 x^8}\)[/tex].
### Conclusion:
Based on the analysis:
- [tex]\(\frac{\sqrt{5 x^7}}{\sqrt{3 x}} \cdot \sqrt{15 x^2}\)[/tex] simplifies to [tex]\(5 x^4\)[/tex].
- [tex]\(\sqrt{25 x^8}\)[/tex] simplifies to [tex]\(5 x^4\)[/tex].
- [tex]\(5 x^4\)[/tex] remains [tex]\(5 x^4\)[/tex].
- [tex]\(5 x^2 \sqrt{x^6}\)[/tex] simplifies to [tex]\(5 x^5\)[/tex].
- [tex]\(25 x^8\)[/tex] remains [tex]\(25 x^8\)[/tex].
The combined, detailed evaluation of the expressions reaffirms that the resulting answers align perfectly.
