Question

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Answer 1

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Answer 2

a + b

and we want to multiply it by:

  a - b

so that the result will be simpler:

 a^2 - b^2

The question is, when will this be simpler? It will if both a and b are  

either radicals or mere numbers. So if you have expressions like:

  2 + 4sqrt(3)    or    5sqrt(3) - 3sqrt(5)

you can simplify them by multiplying them by their conjugates:

  2 - 4sqrt(3)    or    5sqrt(3) + 3sqrt(5)

respectively, so that there will be no radicals left.

If you have more than two terms in an expression, you have to decide  

how to break it up into "a" and "b"; I'm not sure that this is  

technically a conjugate, but it is the same idea extended. In your  

example:

   3 + sqrt(3) - sqrt(6)

there would be no benefit in breaking it up as:

  [ 3 ] + [ sqrt(3) - sqrt(6) ]

since squaring [ sqrt(3) - sqrt(6) ] doesn't simplify it; but we can  

take a step in the right direction by writing it as either:

   [ 3 + sqrt(3) ] - [ sqrt(6) ]

or as:

   [ 3 - sqrt(6) ] + [ sqrt(3) ]

In each case, the left part ("a") will not simplify when you square it,  

but the right side ("b") will. You can then gather terms and use a  

second conjugate to finish the job. Try both ways and see what happens.

Now, if your denominator is sqrt(a+b), it just doesn't have the form  

of a sum, so there is no way to form a conjugate. You could try  

multiplying by sqrt(a-b), of course, but all you will get is  

sqrt(a^2 - b^2), which doesn't help at all. So the best you can do to  

simplify:

      1

  ---------

  sqrt(a+b)

is to multiply numerator and denominator by sqrt(a+b) itself:

      1       sqrt(a+b)

  --------- = ---------

  sqrt(a+b)     (a+b)