1. The problem is to understand why we multiply by the conjugate of the numerator instead of the denominator when simplifying expressions involving radicals. 2. The conjugate of a binomial expression $a + b$ is $a - b$, and vice versa. Multiplying by the conjugate helps eliminate radicals in either the numerator or denominator by using the difference of squares formula: $$ (a+b)(a-b) = a^2 - b^2 $$ 3. When the denominator contains a radical, we multiply numerator and denominator by the conjugate of the denominator to rationalize it, removing the radical from the denominator. 4. However, if the numerator contains radicals and the denominator does not, or if the problem specifically requires simplifying the numerator, we multiply by the conjugate of the numerator. 5. The choice depends on which part contains radicals and which part we want to simplify. Multiplying by the conjugate of the numerator is done to rationalize or simplify the numerator, just as multiplying by the conjugate of the denominator is done to rationalize the denominator. 6. In summary, we multiply by the conjugate of the expression containing radicals that we want to simplify or rationalize, whether numerator or denominator.