# Rationalize a 3 term Denominator

by Asia
(Las Vegas)

Rationalize a 3 term Denominator

Rationalize a Denominator containing 3 terms

The difference of squares formula states that:

(a + b)(a − b) = a^2 − b^2

You can apply the same reasoning to rationalize a denominator which contains three terms by grouping the terms.

Rationalize the denominator of the following expression.

1/(1+3^1/2-5^1/2)

### Comments for Rationalize a 3 term Denominator

 Feb 15, 2011 Rationalize a 3 term Denominator by: Staff The question: by Asia (Las Vegas)1/(1+3^1/2-5^1/2)The answer:Your problem has three terms in the denominator: a + b + cHowever, imagine for a moment how you would rationalize a denominator with only two terms: a + b.As you know, if the denominator contains only two terms, you could rationalize the denominator by multiplying the denominator by its conjugate: a - b. The difference of squares formula states that:(a + b)(a − b) = a^2 − b^2You can apply the same reasoning to rationalize a denominator which contains three terms. GROUP THE TERMS as follows:a + b + c = (a + b) + cThe difference of squares formula states that:[(a + b) + c] * [(a + b) - c] = (a + b) ^2 − c^2Original Problem:1/(1+3^1/2-5^1/2)Group the denominator so that the difference of squares formula can be applied:1/[(1 + 3^1/2) - 5^1/2]Multiply by the conjugate: [(1 + 3^1/2) + 5^1/2]1/[(1 + 3^1/2) - 5^1/2]{1/[(1 + 3^1/2) - 5^1/2]} * {[(1 + 3^1/2) + 5^1/2] / [(1 + 3^1/2) + 5^1/2]}(1 + 3^1/2 + 5^1/2) / [(1 + 3^1/2)^2 + (5^1/2)^2](1 + 3^1/2 + 5^1/2) / (1 + 2*3^1/2 +3 ? 5)(1 + 3^1/2 + 5^1/2) / (2*3^1/2 ? 1)Repeat the process - Multiply by the new conjugate: (2*3^1/2 + 1)(1 + 3^1/2 + 5^1/2) / (2*3^1/2 ? 1)[(1 + 3^1/2 + 5^1/2) / (2*3^1/2 ? 1)] * [(2*3^1/2 + 1)/ (2*3^1/2 + 1)][(1 + 3^1/2 + 5^1/2) * (2*3^1/2 + 1)] / [(2*3^1/2 ? 1) * (2*3^1/2 + 1)][(1 + 3^1/2 + 5^1/2) * (2*3^1/2 + 1)] / (2*3^1/2)^2 ? 1^2)[(1 + 3^1/2 + 5^1/2) * (2*3^1/2 + 1)] / (12 - 1)[(1 + 3^1/2 + 5^1/2) * (2*3^1/2 + 1)] / 11(2 * 3^1/2 + 6 + 2 * 15^1/2 + 1 + 3^1/2 + 5^1/2) / 11(7 + 3 * 3^1/2 + 5^1/2 + 2 * 15^1/2) / 11The final answer is: (7 + 3 * 3^1/2 + 5^1/2 + 2 * 15^1/2) / 11Thanks for writing.Staff www.solving-math-problems.com

 Sep 06, 2016 rationalixe the denominator NEW by: Anonymous i ask rationalize the denominaor with three terms and diffrents coeffients

 Jun 03, 2017 Plz tell how to solve the given ques NEW by: Anonymous √3-√4 upon √3+√4-√7

 Oct 06, 2021 Math Format by: Anonymous I'm not gonna lie, but the format the answer was written in was a royal pain in the a** to read. I couldn't even copy and paste it to WolframAlpha or some other math site to make it legible because the format was either incompatible or the order of operations was incorrect. This is very annoying as this is the only material I could find online regarding rationalizing any denominator with more than two terms.