Conjugate

Conjucates are very useful when it comes to simplifying fractions with an irrational denominater. A conjugate is where you change the signs between two terms. Take a look at the polynomial below:

 1
----
5+√3

If you try to add this fraction to a rational fraction like 1/3, it will be very exhausting to solve it. It will take a very long time as well. Fortunately, there is a very simple solution. That is using the conjugate.

We want to make it where the denominater is a rational number. Then, it could be added, multiplied, etc. To get the conjugate, just change the sign of the denominater. The conjugate of 5+√3 is 5-√3.

Now, if you multiply both sides by the conjugate, you get an irrational answer for the numerator, but a rational outcome for the denominater.

 1     5-√3       5-√3
---- x ---- = ------------
5+√3   5-√3   (5+√3)(5-√3)

When we get the answer, we could still simplify the denominater. There is a rule which will be very helpful:

(a+b)(a-b) = a²-b²

a can be 5, and b can be √3. Then, the denominater will be 25-3=22. Thus, the simplified fraction is the following:

5-√3
----
 22

Using conjugates, we changed a fraction with an irrational denominater to a fraction with a rational denominater.

 1     5-√3
---- = ----
5+√3    22