How Do You Make a Complex Denominator Real in a Fractional Expression?

[kelp]

Hello,
I am trying to simplify a fractional expression with four terms on the bottom, two of the terms are imaginary. How would I go about making the denominator real?
The denominator is as follows:
(1/jwC + R3 + jwL + R2)
The j is the imaginary number. Everything else is a constant. I know with two terms, you can just multiply by the complex conjugate. Not sure how to do it with four terms.

 

[danago]

It is no different with four terms.

The complex conjugate of (R3+R2)+j(wC+wL) is (R3+R2)-j(wC+wL)

 

[Mentallic]

Exactly what danago has said, when you're shown the technique of making the denominator of a complex number a+ib real by multiplying by its complex conjugate a-ib, the a and b can stand for any real number (however complicated).

For example, if $$a=x+y+1$$ and $$b=x^2+y^2+2$$ then the complex number

$$x+y+1+i(x^2+y^2+2)$$ should be multiplied by its complex conjugate $$x+y+1-i(x^2+y^2+2)$$ to obtain $$a^2+b^2$$ or $$(x+y+1)^2+(x^2+y^2+2)^2$$.

Now, the same complex number could have its real and imaginary parts split up, for example the above could be expressed as

$$x(1+ix)+y(1+iy)+2i+1$$

Now in this case it is less evident what the complex conjugate should be. You need to always expand, then collect the real terms and the imaginary terms separately. In other words, expand, then factorize out the i and all those terms that go with it are the imaginary terms while those without an i factorized out are the real terms. (if that makes any sense).

 

[kelp]

Thanks guys!