How to Rationalize the Denominator
2026-02-28 19:03 Diff
https://brightchamps.com/en-us/math/numbers/rationalize-the-denominator

When the denominator has three terms or trinomials, like \(a±\sqrt{b}±\sqrt{c}\), the process of rationalizing is more complex. To rationalize a trinomial denominator, treat two terms as a single binomial, and the third term separately. Select a rationalizing factor that simplifies at least one irrational term when multiplied. If radicals are not eliminated completely after the first rationalization process, repeat the process with the obtained result to eliminate the remaining irrational terms.   


For instance, the given expression is \(\frac{1}{1 + \sqrt{2} - \sqrt{3}} \)


Step 1: Select two terms to form a binomial. Here, we choose \((1 + \sqrt{2}) \) and treat −√3 as the third term. So, the conjugate of the three terms is \((1 − \sqrt{2}) + \sqrt{3}\).


Now, we can multiply this conjugate with both the numerator and the denominator.


\(\frac{1}{1 + \sqrt{2} - \sqrt{3}} \times \frac{1 - \sqrt{2} + \sqrt{3}}{1 - \sqrt{2} + \sqrt{3}} \)

Step 2: Simplify the denominator by using the difference of squares formula: 

(a − b) (a + b) = a2 − b2

Here,\( a = (1 + \sqrt{2})\) and \(b = \sqrt{3}\)

\((1 + \sqrt{2})^2 − (\sqrt{3})^2\)

We can expand it to:

\(1 + 2\sqrt{2} + 2 − 3\)

\(= 3 + 2\sqrt{2} − 3\)

\(= 2\sqrt{2}\)

Here the fraction becomes:

\(\frac{1 + \sqrt{2} + \sqrt{3}}{2 \sqrt{2}} \)

Step 3: Multiply by √2 in both the numerator and the denominator. 

\(\frac{(1 + \sqrt{2} + \sqrt{3}) \cdot 2\sqrt{2} \times \sqrt{2}}{\sqrt{2}} \)

So, we can expand the numerator as:

\(\sqrt{2 }+ 2 + \sqrt{6 }\)

Next, we can expand the denominator as:

\(2\sqrt{2} × \sqrt{2} = 4 \)

Therefore, the final answer is \(\frac{\sqrt{2} + 2 + \sqrt{6}}{4} \).