To simplify the expression [tex]\(\frac{x - \sqrt{y}}{5x - \sqrt{y}}\)[/tex] by rationalizing the denominator, follow these steps:
1. Identify the Expression:
We start with the given expression:
[tex]\[
\frac{x - \sqrt{y}}{5x - \sqrt{y}}
\][/tex]
2. Recognize the Structure:
There is no direct way to rationalize the denominator since it doesn't involve a binomial sum or difference that can be multiplied by a conjugate to remove the square root. However, we observe that the expression itself does not need rationalization in the conventional sense as no complex denominators are present here beyond the square root term.
3. Check for Common Factors:
Next, we look for any common factors in the numerator and the denominator, which might allow us to simplify the expression. In this case, the numerator [tex]\(x - \sqrt{y}\)[/tex] and the denominator [tex]\(5x - \sqrt{y}\)[/tex] do not have any common factors that we can cancel out.
4. Simplify Directly:
Given that there are no common factors and no further simplification that removes the square root, the expression is already simplified.
Thus, the simplified expression remains:
[tex]\[
\frac{x - \sqrt{y}}{5x - \sqrt{y}}
\][/tex]
This is the most simplified form of the given expression.
