1. Let's clarify the concept of conjugates in fractions. 2. The conjugate of a binomial expression like $a + b$ is $a - b$, and vice versa. 3. When you multiply a fraction by the conjugate of its denominator, you multiply both numerator and denominator by that conjugate to rationalize the denominator. 4. This means the conjugate affects both numerator and denominator because you multiply the entire fraction by a form of 1: $$\frac{a}{b} \times \frac{\text{conjugate of } b}{\text{conjugate of } b}$$ 5. The purpose is to eliminate radicals or complex numbers from the denominator. 6. For example, if you have $$\frac{1}{\sqrt{2} + 1}$$, multiply numerator and denominator by $$\sqrt{2} - 1$$: $$\frac{1}{\sqrt{2} + 1} \times \frac{\sqrt{2} - 1}{\sqrt{2} - 1} = \frac{\sqrt{2} - 1}{(\sqrt{2} + 1)(\sqrt{2} - 1)}$$ 7. The denominator simplifies using the difference of squares formula: $$ (\sqrt{2})^2 - 1^2 = 2 - 1 = 1 $$ 8. So the fraction becomes: $$ \frac{\sqrt{2} - 1}{1} = \sqrt{2} - 1 $$ 9. Notice the conjugate multiplied both numerator and denominator, not just the numerator. 10. Therefore, the conjugate affects the entire fraction, not only the numerator.