Finding a common demominator

[rtareen]

Hello, can you please show me the steps to get from equation 22-6 to equation 22-7? Please show all steps in between, making them as little as possible (dont combine too any steps into a single step). Thanks!

 

[Steven G]

We do not do step by step here as that will be doing the work for you. This is a math help site! We expect you to do the work with our help.

Let's just talk about the two fractions being subtracted inside the big parenthesis. What is the common denominator? What is (1 - d/2)*(1+d/2)? So what will (1-d/2)²*(1+d/2)² equal?

 

[rtareen]

Ok. (1-d/2)(1+d/2) = 1 - (d^2)/4
And the square of that is (1 - (d^2)/4)^2 = (1 - (d/2)^2)^2
I think thats right. Its pretty close to the denomiator they get. We just didnt include the z.
Its been so long since ive done common denominators

 

[Steven G]

Yes, I missed the z. So can you please redo the multiplication the way I meant to ask it and see what you get after subtracting the two fractions.

 

[rtareen]

ok. Since the fractions are so big ill do the numerator and denominator seperately.
first the common denominator will be
denominator = ((1-d/2z)(1+d/2z))^2 = (1 - (d/2z)^2)^2, which is the denominator we want.

The numerator on the left becomes (1 + d/2z)^2 = 1 + d/z + d^2/4z^2
The numerator on the right becomes (1 - d/2z)^2 = 1 - d/z + d^2/4z^2

Now since there is a common denominator we can subtract the numerators.

numerator = 1 + d/z + d^2/4x^2 - 1 + d/z - d^2/4z^2 = 2d/z

numerator/ denominator = (2d/z)/((1 - (d/2z)^2)^2)