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(4 - 2sqrt(3) ) - Rationalize the Denominator











































Rationalize the Denominator

(3 + 5√(3)) / (4 - 2√(3))


REMOVE the √(3) from the denominator by using the DIFFERENCE OF two SQUARES formula.

(a + b)(a − b) = a^2 − b^2

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Jan 24, 2011
Rationalize the Denominator
by: Staff


The question:

(3 + 5ã3) / (4 - 2ã3)


The answer:

(3 + 5ã3) / (4 - 2ã3)


Your mathematical notation did not display properly. I assume you actually mean:

[3 + 5sqrt(3)] / [4 – 2sqrt(3)]


As you already know, rationalizing the denominator means to remove all radicals from the denominator.

We are going to REMOVE the SQRT(3) from the denominator USING the DIFFERENCE OF two SQUARES formula (also called the difference of perfect squares).

The difference of squares formula states that:

(a + b)(a − b) = a^2 − b^2



IF WE COULD APPLY THE DIFFERENCE OF SQUARES to the denominator by MULTIPLYING THE DENOMINATOR BY ITS CONJUGATE, the radical in the denominator would be eliminated. As a matter of fact, the complicated denominator would be transformed into the number 4.


The CONJUGATE of [4 – 2sqrt (3)] is [4 + 2sqrt(3)].

[4 – 2sqrt(3)]*[4 + 2sqrt(3)] = (4)^2 – (2sqrt(3))^2

(4)^2 – (2sqrt(3))^2 = 4*4 – (2sqrt(3))* (2sqrt(3))

4*4 – (2sqrt(3))* (2sqrt(3))

16 – 2*2*sqrt(3)*sqrt(3)

16 – 4*3

16 – 12

4

If we could . . . that is our goal.

The question is how can we accomplish this without changing the value of the original fraction?


This is how.

The denominator can be multiplied by any number (or expression) as long as the numerator is multiplied by the same thing.

Multiplying both the numerator and denominator by the same number (or the same expression) is the same as multiplying the original fraction by a new fraction whose value is 1. (any number)/(the same number) = 1. Multiplying by 1 does not change the value of the original fraction.


To apply the difference of squares formula to your problem: multiply both the numerator and denominator by [4 + 2sqrt(3)]


[3 + 5sqrt(3)] / [4 – 2sqrt(3)]


{[3 + 5sqrt(3)] / [4 – 2sqrt(3)]}*1

{[3 + 5sqrt(3)] / [4 – 2sqrt(3)]}*{[4 + 2sqrt(3)] / [4 + 2sqrt(3)]}

{[3 + 5sqrt(3)] )*[4 + 2sqrt(3)]} / {[4 – 2sqrt(3)]* [4 + 2sqrt(3)]}

{[3 + 5sqrt(3)] )*[4 + 2sqrt(3)]} / 4

[42 + 26sqrt(3)] / 4


The final answer to your question is: [42 + 26sqrt(3)] / 4





Thanks for writing.


Staff
www.solving-math-problems.com


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