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Rationalize a 3 term Denominator

by Asia
(Las Vegas)

Rationalize a 3 term Denominator

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)

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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 + 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


Original 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) / 11


The final answer is: (7 + 3 * 3^1/2 + 5^1/2 + 2 * 15^1/2) / 11



Thanks 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

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