Answer :
Let's simplify the given expression step-by-step to determine which choice is equivalent to the expression when [tex]\( x \geq 0 \)[/tex].
The given expression is:
[tex]\[ \sqrt{50 x^3}-\sqrt{25 x^3}+5 \sqrt{x^3}-\sqrt{2 x^3} \][/tex]
### Simplifying Each Term
1. First Term: [tex]\( \sqrt{50 x^3} \)[/tex]
[tex]\[ \sqrt{50 x^3} = \sqrt{50} \cdot \sqrt{x^3} = \sqrt{25 \cdot 2} \cdot \sqrt{x^3} = 5\sqrt{2} \cdot x^{3/2} \][/tex]
2. Second Term: [tex]\( \sqrt{25 x^3} \)[/tex]
[tex]\[ \sqrt{25 x^3} = \sqrt{25} \cdot \sqrt{x^3} = 5 \cdot x^{3/2} \][/tex]
3. Third Term: [tex]\( 5 \sqrt{x^3} \)[/tex]
[tex]\[ 5 \sqrt{x^3} = 5 \cdot x^{3/2} \][/tex]
4. Fourth Term: [tex]\( \sqrt{2 x^3} \)[/tex]
[tex]\[ \sqrt{2 x^3} = \sqrt{2} \cdot \sqrt{x^3} = \sqrt{2} \cdot x^{3/2} \][/tex]
### Combine the Simplified Terms
Now we add and subtract the simplified terms:
[tex]\[ 5 \sqrt{2} \cdot x^{3/2} - 5 \cdot x^{3/2} + 5 \cdot x^{3/2} - \sqrt{2} \cdot x^{3/2} \][/tex]
Combine like terms:
[tex]\[ (5 \sqrt{2} \cdot x^{3/2}) - (5 \cdot x^{3/2}) + (5 \cdot x^{3/2}) - (\sqrt{2} \cdot x^{3/2}) \][/tex]
The terms [tex]\(- 5 \cdot x^{3/2}\)[/tex] and [tex]\(+ 5 \cdot x^{3/2}\)[/tex] cancel each other out, leaving:
[tex]\[ 5 \sqrt{2} \cdot x^{3/2} - \sqrt{2} \cdot x^{3/2} \][/tex]
Factor out [tex]\( \sqrt{2} \cdot x^{3/2} \)[/tex]:
[tex]\[ (5 - 1) \cdot \sqrt{2} \cdot x^{3/2} = 4 \cdot \sqrt{2} \cdot x^{3/2} \][/tex]
### Comparing With Choices
- Choice A: [tex]\(4 \sqrt{x} \)[/tex] does not match the expression.
- Choice B: [tex]\(28 \sqrt{x^3}\)[/tex] does not match the expression.
- Choice C: [tex]\(5 \sqrt{2 x} \)[/tex] does not match the expression.
- Choice D: [tex]\(4 x \sqrt{2 x} \)[/tex] does not match the expression.
After detailed step-by-step evaluation, it appears that none of the options provided match the simplified expression [tex]\(4 \cdot \sqrt{2} \cdot x^{3/2}\)[/tex].
Therefore, the answer to the given problem is:
```
None
```
The given expression is:
[tex]\[ \sqrt{50 x^3}-\sqrt{25 x^3}+5 \sqrt{x^3}-\sqrt{2 x^3} \][/tex]
### Simplifying Each Term
1. First Term: [tex]\( \sqrt{50 x^3} \)[/tex]
[tex]\[ \sqrt{50 x^3} = \sqrt{50} \cdot \sqrt{x^3} = \sqrt{25 \cdot 2} \cdot \sqrt{x^3} = 5\sqrt{2} \cdot x^{3/2} \][/tex]
2. Second Term: [tex]\( \sqrt{25 x^3} \)[/tex]
[tex]\[ \sqrt{25 x^3} = \sqrt{25} \cdot \sqrt{x^3} = 5 \cdot x^{3/2} \][/tex]
3. Third Term: [tex]\( 5 \sqrt{x^3} \)[/tex]
[tex]\[ 5 \sqrt{x^3} = 5 \cdot x^{3/2} \][/tex]
4. Fourth Term: [tex]\( \sqrt{2 x^3} \)[/tex]
[tex]\[ \sqrt{2 x^3} = \sqrt{2} \cdot \sqrt{x^3} = \sqrt{2} \cdot x^{3/2} \][/tex]
### Combine the Simplified Terms
Now we add and subtract the simplified terms:
[tex]\[ 5 \sqrt{2} \cdot x^{3/2} - 5 \cdot x^{3/2} + 5 \cdot x^{3/2} - \sqrt{2} \cdot x^{3/2} \][/tex]
Combine like terms:
[tex]\[ (5 \sqrt{2} \cdot x^{3/2}) - (5 \cdot x^{3/2}) + (5 \cdot x^{3/2}) - (\sqrt{2} \cdot x^{3/2}) \][/tex]
The terms [tex]\(- 5 \cdot x^{3/2}\)[/tex] and [tex]\(+ 5 \cdot x^{3/2}\)[/tex] cancel each other out, leaving:
[tex]\[ 5 \sqrt{2} \cdot x^{3/2} - \sqrt{2} \cdot x^{3/2} \][/tex]
Factor out [tex]\( \sqrt{2} \cdot x^{3/2} \)[/tex]:
[tex]\[ (5 - 1) \cdot \sqrt{2} \cdot x^{3/2} = 4 \cdot \sqrt{2} \cdot x^{3/2} \][/tex]
### Comparing With Choices
- Choice A: [tex]\(4 \sqrt{x} \)[/tex] does not match the expression.
- Choice B: [tex]\(28 \sqrt{x^3}\)[/tex] does not match the expression.
- Choice C: [tex]\(5 \sqrt{2 x} \)[/tex] does not match the expression.
- Choice D: [tex]\(4 x \sqrt{2 x} \)[/tex] does not match the expression.
After detailed step-by-step evaluation, it appears that none of the options provided match the simplified expression [tex]\(4 \cdot \sqrt{2} \cdot x^{3/2}\)[/tex].
Therefore, the answer to the given problem is:
```
None
```
