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Rationalizing the denominator

by Emilee
(USA)











































5/((6 + 3√2) - (2 + 2√2))

Rationalize the denominator of 5 over open parentheses 6 plus 3 square root of 2 close parentheses minus open parentheses 2 plus 2 square root of 2 close parentheses:

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Jan 09, 2013
Remove Radicals from Denominator
by: Staff


Answer


Part I

Math – rationalize the denominator in 6 steps







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”.

The technique used to rationalize the denominator of a fraction is always the same:

Multiply the original fraction (which contains radicals in the denominator) by another fraction (a / a).

To ensure the value of the original fraction (which contains radicals in the denominator) will not change, the second fraction (a / a) will always to equal to 1. The value of the numerator (a) in the second fraction and the value of the denominator (a) in the second fraction will always be the same.

The denominator of the new fraction (after multiplication) will contain no radicals.

Rationalizing the denominator of any fraction is really a problem of determining what the second fraction should be, so that:

= [original fraction] * 1

a = currently unknown expression

= [(original numerator) / (original denominator)] * (a / a)

= [(original numerator) * a] / [(original denominator) * a]

After multiplication is complete, the denominator of the new fraction [(original denominator) * a] will not contain any radicals.

[(original denominator) * a] = a value which does not contain radicals



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Jan 09, 2013
Remove Radicals from Denominator
by: Staff


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Part II

Your problem: rationalize the following fraction:


5/[(6 + 3√2) - (2 + 2√2)]


Combine like terms in the denominator

= 5/(6 + 3√2 - 2 - 2√2)

= 5/(6 - 2 + 3√2 - 2√2)

= 5/(4 + √2)



Your problem has two terms in the denominator: a + b

You can rationalize the denominator by applying the Difference of Squares formula.


The difference of squares formula states that:

(a + b)(a - b) = a² - b²


You can remove the radicals from the denominator in your problem by multiplying the denominator by its conjugate: a - b.


“IF” the original denominator (4 + √2) could be multiplied by its conjugate (4 - √2), then the √ sign in the denominator will disappear, since:

(4 + √2) * (4 - √2)

= 4² - (√2)²

= 16 - 2

= 14

However, in order to preserve the value of the original fraction, both the numerator and denominator must each be multiplied by the same amount: (4 - √2).

To apply this concept, multiply the original fraction by [(4 - √2)] / [(4 - √2)]. The fraction [(4 - √2)] / [(4 - √2)] is equal to 1, so the original fraction is merely being multiplied by 1. As you can see by the following illustration, the value of the original fraction has not been changed.


= [original fraction]

= [original fraction] * [(4 - √2)] / [(4 - √2)]

= [original fraction] * 1

= [original fraction]


Therefore,

= [original fraction] * [(4 - √2)] / [(4 - √2)]

= [5/(4 + √2)] * [(4 - √2)] / [(4 - √2)]


Multiply both numerators and multiply both denominators, just as you would when multiplying any two fractions:


= [5 * (4 - √2)] / [(4 + √2) * (4 - √2)]

= [20 - 5√2] / [(4² - (√2)²]

= (20 - 5√2) / (16 - 2)

= (20 - 5√2) / 14


( ≈ 0.9234951562953)




>>>The final answer is: (20 - 5√2) / 14




Check this answer against the original expression with a calculator:

Final answer:

(20 - 5√2) / 14 ≈ 0.9234951562953


Original expression:

5/((6 + 3√2) - (2 + 2√2)) ≈ 0.9234951562953


Since the decimal value computed from the original expression and the decimal value computed from the final answer are the same, the solution is valid.






Thanks for writing.

Staff
www.solving-math-problems.com



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