rationalizing the denominator

Rationalizing the denominator means rewriting a fraction so that no radical is left on the bottom, without changing the fraction's value. The trick is to multiply the top and bottom by a carefully chosen factor that clears the root downstairs.

For a single square root, multiply by that same root over itself. To rationalize 1/sqrt(2), multiply top and bottom by sqrt(2): 1/sqrt(2) = (1 × sqrt(2)) / (sqrt(2) × sqrt(2)) = sqrt(2)/2. Since you multiplied by sqrt(2)/sqrt(2) = 1, the value is unchanged, but the denominator is now the whole number 2.

When the denominator is a sum or difference involving a root, like 1/(3 + sqrt(2)), you multiply by the conjugate (3 − sqrt(2)) over itself; the difference-of-squares pattern wipes out the radical. The custom dates from hand computation, where dividing by a whole number was far easier than dividing by an endless decimal; it also gives a single standard form so answers are easy to compare.