Sure! Let's go through the problem step-by-step to determine which choice is equivalent to the given quotient when [tex]\( x \geq 0 \)[/tex]:
We start with the given quotient:
[tex]\[ \frac{\sqrt{18x}}{\sqrt{50}} \][/tex]
Step 1: Simplify both the numerator and the denominator.
First, we simplify the numerator [tex]\(\sqrt{18x}\)[/tex]:
- Notice that [tex]\(\sqrt{18x} = \sqrt{18} \cdot \sqrt{x}\)[/tex].
- We can further simplify [tex]\(\sqrt{18}\)[/tex] by writing it as [tex]\(\sqrt{9 \cdot 2} = \sqrt{9} \cdot \sqrt{2} = 3\sqrt{2}\)[/tex].
- Thus, [tex]\(\sqrt{18x} = 3\sqrt{2}\sqrt{x}\)[/tex].
Next, simplify the denominator [tex]\(\sqrt{50}\)[/tex]:
- Notice that [tex]\(\sqrt{50} = \sqrt{25 \cdot 2} = \sqrt{25} \cdot \sqrt{2} = 5\sqrt{2}\)[/tex].
So, our quotient now looks like this:
[tex]\[ \frac{3\sqrt{2} \sqrt{x}}{5\sqrt{2}} \][/tex]
Step 2: Cancel out common terms in the numerator and the denominator.
- We notice that both the numerator and the denominator have a common factor of [tex]\(\sqrt{2}\)[/tex].
- By canceling [tex]\(\sqrt{2}\)[/tex] from both, we get:
[tex]\[ \frac{3\sqrt{x}}{5} \][/tex]
Step 3: Compare with the given answer choices.
Looking at the simplified expression [tex]\( \frac{3\sqrt{x}}{5} \)[/tex], we see that it matches choice C.
Therefore, the equivalent expression to the given quotient is:
[tex]\[ \boxed{\frac{3\sqrt{x}}{5}} \][/tex]
So, the correct choice is:
C. [tex]\(\frac{3 \sqrt{x}}{5}\)[/tex]
