Rationalise the Denominator problem: Simplify 2/((√6) + 2)

What you want to do is multiply both the numerator and the denominator by the conjugate. To find the conjugate, look for a factor that will remove any roots in the denominator. The way I'd go about finding it is as follows:

The denominator is of the form (a + b). If I multiply by (a - b), I have (a+b)(a-b). After factoring, I have a^2 - ab + ab - b^2. The -ab and +ab cancel out, leaving only a^2 - b^2. Since a in this problem is 6\displaystyle \sqrt{6} and b is 2, a^2 is just 6 and b^2 is 4. So, a^2 - b^2 is 2. Thus, the conjugate is 6−2\displaystyle \sqrt{6} - 2.

Multiply the numerator and denominator by that conjugate and the fraction becomes:

26+2⋅6−26−2=2⋅(6−2)2\displaystyle \frac{2}{\sqrt{6}+2}\cdot \frac{\sqrt{6}-2}{\sqrt{6}-2} = \frac{2\cdot \left(\sqrt{6}-2\right)}{2}. There is a 2 in the numerator and 2 in the denominator, so those cancel. And what remains is the answer you were given of 6−2\displaystyle \sqrt{6}-2

I want to point out that although the answer to this problem happens to be the same as the conjugate, it's just a coincidence. In most problems of this sort, that won't be the case.
 
I see the error I was making now. I was assuming the denominator was 6 - 2, giving 4, rather than 6 - 2(squared), giving 2.
Many thanks ksdhart.
 
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