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Which choice is equivalent to the expression below when [tex][tex]$x \geq 0$[/tex][/tex]?

[tex]\sqrt{50 x^3} - \sqrt{25 x^3} + 5 \sqrt{x^3} - \sqrt{2 x^3}[/tex]

A. [tex]4 \sqrt{x}[/tex]

B. [tex]28 \sqrt{x^3}[/tex]

C. [tex]5 \sqrt{2 x}[/tex]

D. [tex]4 x \sqrt{2 x}[/tex]


Sagot :

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
```