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7/(7 - √7) - Rationalize the Denominator











































7/(7 - √7)

Rationalize the Denominator to simplify the fraction.

Rationalizing the Denominator will remove all radicals from the denominator.

Comments for 7/(7 - √7) - Rationalize the Denominator

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Apr 21, 2011
Rationalize the Denominator
by: Staff


The question:

rationalize the denominator; 7/(7-square rt7)


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:

7/(7 - √7)


“IF” the denominator could by multiplied by its conjugate [which is (7 + √7)], then the √ sign in the denominator would disappear, since:

(7 - √7) * (7 + √7) = 7² - (√7)² = 49 – 7 = 42


However, in order to preserve the value of the original fraction, both the numerator and denominator must each be multiplied by the same amount: (7 + √7).

To apply this concept, multiply the original fraction by (7 + √7)/(7 + √7). The fraction (7 + √7)/(7 + √7) 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] * [(7 + √7)/(7 + √7)]

= [original fraction] * 1

= [original fraction]


Therefore,

= [original fraction] * [(7 + √7)/(7 + √7)]

= [7/(7 - √7)] * [(7 + √7)/(7 + √7)]

= 7*(7 + √7) / [(7 - √7) * (7 + √7)]

= 7 * (7 + √7) / [7² - (√7)²]

= 7 * (7 + √7) / (49 – 7)

= 7 * (7 + √7) / (42)


Reduce this fraction to its lowest terms

= 7 * (7 + √7) / (6 * 7)

= [(7 + √7) / 6 ] * (7 / 7)

= [(7 + √7) / 6 ] * (1)

= (7 + √7) / 6


The final answer is:

7/(7 - √7) = (7 + √7) / 6



Check the results:

Using a calculator, compare the numerical results of the original fraction and the final answer:

Original fraction

7 / (7 - sqrt(7)) = 1.60762522

Final answer

(7 + sqrt(7)) / 6 = 1.60762522


Since the numerical values of the original fraction and the final answer are the same, the final answer is correct.



Thanks for writing.


Staff
www.solving-math-problems.com


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