Rationalizing Denominators in Radicals Sometimes we would like a radical expression to be written in such a way that there is no radical in the denominator. To achieve this goal we use a process called rationalizing the denominator.

Example 1 Rationalize the denominator: We would like to eliminate the radical from the denominator. Multiply both numerator and denominator by

Notice that the radical in the denominator has been eliminated. Note also that we multiplied by an expression equivalent to 1, so the value of the expression was not changed.

Example 2 Rationalize the denominator: We would like to eliminate the radical from the denominator. Since the radical in the denominator is a fourth root, we need a perfect fourth power under the radical.

Consider this method to determine the necessary radicand. Since there is one factor of 3 already … … three more factors of 3, would give a total of four, and thus a perfect fourth power.

Consider this method to determine the necessary radicand. Since there is one factor of 3 in the radicand already … … three more factors of 3, would give a total of four, and thus a perfect fourth power.

Example 3 Rationalize the denominator: We would like to eliminate the radical from the denominator. Since the radical in the denominator is a cube root, we need a perfect third power under the radical.

One factor of 5 already … … so two more factors of 5 are needed. Two factors of y already … … so one more factor of y is needed.

One factor of 5 already … … so two more factors of 5 are needed. Two factors of y already … … so one more factor of y is needed.