Conjugate

Property The conjugate of an irrational number in the form $$ a + \sqrt{b} $$ is $$ a \sqrt{b} $$. Explanation When the denominator is a binomial team like $$ (2 + \sqrt{3}) $$, you need its secret twin—the conjugate! Multiplying by the conjugate uses the difference of squares pattern to magically eliminate the radical from the denominator, making your expression neat and tidy. Examples $$ \frac{6}{4 + \sqrt{2}} = \frac{6}{4 + \sqrt{2}} \cdot \frac{4 \sqrt{2}}{4 \sqrt{2}} = \frac{24 6\sqrt{2}}{16 2} = \frac{12 3\sqrt{2}}{7} $$ $$ \frac{8}{\sqrt{7} 1} = \frac{8}{\sqrt{7} 1} \cdot \frac{\sqrt{7} + 1}{\sqrt{7} + 1} = \frac{8\sqrt{7} + 8}{7 1} = \frac{4\sqrt{7} + 4}{3} $$.