Which is the correct way to rationalize the denominator of an expression with a surd?

To rationalize the denominator of an expression containing a surd, multiplying by the conjugate of the denominator is the most effective method. The conjugate of a binomial expression, such as ( a + \sqrt{b} ), is ( a - \sqrt{b} ). When you multiply a binomial by its conjugate, you utilize the difference of squares identity, which results in the cancellation of the surd in the denominator.

For example, if you have a fraction like ( \frac{1}{\sqrt{a} + b} ), multiplying the numerator and denominator by ( \sqrt{a} - b ) simplifies the denominator to ( (\sqrt{a})^2 - b^2 ), thus eliminating the square root. This approach leaves you with a rational number in the denominator, making the expression easier to work with.

Other options do not effectively rationalize a denominator with a surd. Factoring the denominator completely may not lead to a rational denominator if it still contains surds. Converting the denominator to a perfect square is not a general approach applicable in all cases, as not all denominators can be converted this way without additional context. Taking the square root of the denominator is also