Rationalize the denominator for [tex]$\frac{\sqrt{5}}{\sqrt{2 x}}$[/tex].

A. [tex]x \frac{\sqrt{5}}{2}[/tex]
B. [tex]\frac{\sqrt{5 x}}{2}[/tex]
C. [tex]\frac{\sqrt{10 x}}{2 x}[/tex]
D. [tex]\frac{\sqrt{10 x}}{2}[/tex]



Answer :

To rationalize the denominator of the expression [tex]\(\frac{\sqrt{5}}{\sqrt{2x}}\)[/tex], we need to remove the square root from the denominator. Here are the detailed steps:

1. Rewrite the Expression:
[tex]\[ \frac{\sqrt{5}}{\sqrt{2x}} \][/tex]

2. Multiply Numerator and Denominator by [tex]\(\sqrt{2x}\)[/tex]:
To rationalize the denominator, we multiply both the numerator and the denominator by [tex]\(\sqrt{2x}\)[/tex]:
[tex]\[ \frac{\sqrt{5} \cdot \sqrt{2x}}{\sqrt{2x} \cdot \sqrt{2x}} \][/tex]

3. Simplify the Denominator:
The product of [tex]\(\sqrt{2x} \cdot \sqrt{2x}\)[/tex] is [tex]\(2x\)[/tex]:
[tex]\[ \sqrt{2x} \cdot \sqrt{2x} = 2x \][/tex]

4. Simplify the Numerator:
The product of [tex]\(\sqrt{5} \cdot \sqrt{2x}\)[/tex] is [tex]\(\sqrt{10x}\)[/tex]:
[tex]\[ \sqrt{5} \cdot \sqrt{2x} = \sqrt{10x} \][/tex]

5. Combine the Results:
Therefore, the expression simplifies to:
[tex]\[ \frac{\sqrt{10x}}{2x} \][/tex]

So, the rationalized form of the given expression [tex]\(\frac{\sqrt{5}}{\sqrt{2x}}\)[/tex] is [tex]\(\frac{\sqrt{10x}}{2x}\)[/tex].

Answer (C) [tex]\( \frac{\sqrt{10x}}{2x} \)[/tex]