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]
