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If you are asked to write the reciprocal of a set of terms containing a radical, you will need to rationalize before simplifying. Use the method for monomial or binomial denominators, depending on whichever applies to the problem. 23{\displaystyle 2-{\sqrt {3}}} A reciprocal is created when you invert the fraction. Our expression 23{\displaystyle 2-{\sqrt {3}}} is actually a fraction. It's just being divided by 1. 123{\displaystyle {
rac {1}{2-{\sqrt {3}}}}} Remember, you're actually multiplying by 1, so you have to multiply both the numerator and denominator. Our example is a binomial, so multiply the top and bottom by the conjugate. 1232+32+3{\displaystyle {
rac {1}{2-{\sqrt {3}}}}\cdot {
rac {2+{\sqrt {3}}}{2+{\sqrt {3}}}}} 1232+32+3=2+343=2+3{\displaystyle {
rac {1}{2-{\sqrt {3}}}}\cdot {
rac {2+{\sqrt {3}}}{2+{\sqrt {3}}}}={
rac {2+{\sqrt {3}}}{4-3}}=2+{\sqrt {3}}} Do not be thrown off by the fact that the reciprocal is the conjugate. This is just a coincidence.
Examine the problem. Write the reciprocal as it would usually appear. Multiply by something that can get rid of the radical on the bottom. Simplify as needed.