# Math .. Rationalize the Denominator

Rationalize a denominator containing three terms.

You can apply the same reasoning to rationalize a denominator containing three terms as you would use to rationalize a denominator containing only two terms.

The key is to group the terms.

### Comments for Math .. Rationalize the Denominator

 Jun 20, 2011 Rationalize a 3 term Denominator by: Staff ------------------------------------------------------------------ Part II Repeat the process - Multiply by a fraction containing the new conjugate: (2*3^1/2 + 1) in BOTH the numerator and denominator. (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)] As before, multiply both numerators and multiply both denominators, just as you would when multiplying any two fractions: = [(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 - 1²) [(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) / 11 The final answer is: (7 + 3 * 3^1/2 + 5^1/2 + 2 * 15^1/2) / 11 Check this answer against the original expression with a calculator: Final answer: (7 + 3 * 3^1/2 + 5^1/2 + 2 * 15^1/2) / 11 = 2.0162 Original expression: 1/(1+3^1/2-5^1/2) = 2.0162 Thanks for writing. Staff www.solving-math-problems.com

 Jun 20, 2011 Rationalize a 3 term Denominator by: Staff Part I The question: 1/1+rad3-rad5 The answer: Rationalizing the Denominator is the standard way of simplifying fractions containing radicals in the denominator. Rationalizing the denominator means to “rewrite the fraction so there are no radicals in the denominator”. Your problem as it now stands: 1/(1+3^1/2-5^1/2) Your problem has three terms in the denominator: a + b + c However, 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^2 You can apply the same reasoning to rationalize a denominator which contains three terms. GROUP THE TERMS as follows: a + b + c = (a + b) + c The difference of squares formula states that: [(a + b) + c] * [(a + b) - c] = (a + b) ^2 − c^2 Group the denominator so that the difference of squares formula can be applied: 1/[(1 + 3^1/2) - 5^1/2] Multiply the original fraction by a fraction containing the conjugate: [(1 + 3^1/2) + 5^1/2] in BOTH the numerator and denominator. The fraction [(1 + 3^1/2) + 5^1/2] / [(1 + 3^1/2) + 5^1/2] is equal to 1, so the original fraction is merely being multiplied by 1. As you can see by the following illustration, its value has not been changed. = [original fraction] = [original fraction] * [(1 + 3^1/2) + 5^1/2] / [(1 + 3^1/2) + 5^1/2] = [original fraction] * 1 = [original fraction] Therefore, = [original fraction] * [(1 + 3^1/2) + 5^1/2] / [(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]} Multiply both numerators and multiply both denominators, just as you would when multiplying any two fractions: 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 * [(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) ------------------------------------------------------------------