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-3/the square root of 6, Rationalize the Denominator


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-3/the square root of 6. I know that it multiplies out to -3 square root of 6/6, but do you reduce it to-square root of 6/2 or leave it as -3square root of 6/6? Thanks.

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Nov 16, 2010
Rationalizing the Denominator
by: Staff


The question:

Rationalize the denominator for the following fraction:

-3/the square root of 6.

I know that it multiplies out to -3 square root of 6/6.

Do you reduce it to-square root of 6/2 or leave it as -3square root of 6/6?

Thanks.



The answer:


-3/sqrt(6)

We are not going to change the value of -3/sqrt(6). We are merely rearranging it so that there will be no sqrt in the denominator.

To rationalize the denominator, multiply the fraction -3/sqrt(6) by another fraction: sqrt(6)/sqrt(6).

[Note that the second fraction sqrt(6)/sqrt(6) = 1, so you are really multiplying by 1. Multiplying -3/sqrt(6) by 1 does not change its value.

Why choose the fraction sqrt(6)/sqrt(6)? Why not some other fraction?

I deliberately chose to put the sqrt(6) in both the numerator and denominator because I know that sqrt(6) * sqrt(6) will equal the rational number 6]


[-3/sqrt(6)] * [sqrt(6)/sqrt(6)]

= [(-3) * sqrt(6)] /[sqrt(6) * sqrt(6)]



= -3sqrt(6) /6

Without reducing the fraction any further, the answer is: -3sqrt(6) /6


In general, it’s a good idea to check your final answer with a calculator when you can.

Google’s free on-line calculator is located here:

http://www.google.com/intl/en/help/features.html#calculator


Check the numerical result for your original problem:

Just enter -3/sqrt(6), then press carriage return

According to Google: (-3) / sqrt(6) = -1.22474487

Check the numerical result for your version with the rationalized denominator:

Just enter -3sqrt(6) /6, then press carriage return

According to Google: ((-3) sqrt(6)) / 6 = -1.22474487

Notice that the numerical results for both versions of the expression are the same. That means that we rationalized the denominator correctly.

-3/sqrt(6) = -3sqrt(6) /6 , exactly

The answer to your second question is Yes, you can reduce the answer to a simpler form.

Begin by factoring the denominator:
-3sqrt(6) /6 = [-3 * sqrt(6)] /[2 * 3]

The 3 in the numerator cancels the 3 in the denominator.

[-3 * sqrt(6)] /[2 * 3] = -1 * sqrt(6)] /2
-1 * sqrt(6)] /2 = -sqrt(6)/2

The final answer to your problem is: -sqrt(6)/2





Thanks for writing.


Staff
www.solving-math-problems.com



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