2. What is a Conjugate? Conjugates are pairs of binomials involving radicals that, when multiplied together, become rational (the radicals disappear). Pairs of conjugates are in the form a√b + c√d and a√b - c√d.

3. Why Do the Radicals Disappear? Recall the difference of squares formula: x 2 – y 2 = (x + y)(x – y) Notice that the product of two conjugates can be expressed as (a√b + c√d)(a√b - c√d) Which is in the same form as the right side of the difference of squares formula. Thus, if we expand this product, we get (a√b) 2 – (c√d) 2 = a 2 b – c 2 d Which contains no radicals.

4. How is this Useful? Recall that we want to rationalize the denominator of fractions in order to make them easier to work with. Conjugates give us a way to rationalize fractions that have radicals in binomials in their denominators. By multiplying the numerator and denominator of a fraction by the conjugate of the denominator, we rationalize the denominator without changing the value of the fraction.

5. Example Say we want to rationalize If we multiply the top and bottom of the equation by we can rationalize the denominator without changing the value of the fraction. = 25 – 2 = 23. Thus, we’re left with There are still radicals in the numerator, but that’s okay.

6. Try on your own 1. What is the conjugate of ? 2. Rationalize 3. Rationalize

7. Answers 1. 2. 3.