Step-By-Step Rationalization
"Rationalizing" the denominator of a fraction means applying the invariant property of fractions in order to obtain an equivalent fraction that does not contain radicals in the denominator.
Note
Invariant property of fractions: if you multiply or divide both the numerator and the denominator of a fraction by the same nonzero number, you obtain a fraction equivalent to the original.
In fact, a fraction with the same numerator and denominator equals 1, and multiplying any fraction by 1 does not change its value.
We will call the term used in both the numerator and denominator for the purpose of rationalization the rationalizing factor.
The Denominator Is a Square Root
To rationalize , multiply both the numerator and the denominator by the rationalizing factor .
Therefore: .
Test Yourself
Try to rationalize the fraction given below, then use the app to check whether your solution is correct.
The Denominator Is a Higher Order Root
To rationalize , multiply both the numerator and denominator by the rationalizing factor .
Therefore .
The formula above may look complicated, but it isn't.
The rationalizing factor is a root with the same index as the given one, the same base as the radicand, and an exponent equal to the number of powers needed for the current exponent to match the index of the root.
Test Yourself
Try to rationalize the fraction given below, then use the app to check whether your solution is correct.
The Denominator Is an Algebraic Sum of Two Terms, and One of Them Is a Square Root
To rationalize multiply both the numerator and the denominator by the rationalizing factor .
Therefore: .
Test Yourself
Try to rationalize the fraction given below, then use the app to check whether your solution is correct.