Can someone show me how this is solved algebraically

[chris_0101]

Homework Statement

I need to show how the first equation gives the second equation algebraically

Homework Equations

I've tried multiple methods such as rationalizing the denominator with no avail.

The Attempt at a Solution

Any help will be greatly appreciated,

Thanks.

 

[STEMucator]

Homework Statement

I need to show how the first equation gives the second equation algebraically
View attachment 51672

Homework Equations

I've tried multiple methods such as rationalizing the denominator with no avail.

The Attempt at a Solution

Any help will be greatly appreciated,

Thanks.

Multiply out the numerator of that mess over c^2. Then find a quick common divisor. Then use the rules :

\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}

and

\frac{\frac{a}{b}}{\frac{c}{d}} = \frac{ad}{bc}

 

[chris_0101]

I'm not sure what you mean by "multiply out the numerator of that mess over c^2"?

 

[STEMucator]

I'm not sure what you mean by "multiply out the numerator of that mess over c^2"?

The quantity (\frac{u'+v}{1+vu'/c^2})^2

 

[chris_0101]

yea, but how does one multiply out the numerator?
do you mean like this:
(a/b)/c = (a/b)*(1/c)

 

[STEMucator]

yea, but how does one multiply out the numerator?
do you mean like this:
(a/b)/c = (a/b)*(1/c)

...

(\frac{u'+v}{1+vu'/c^2})^2 = (\frac{u'+v}{1+vu'/c^2})(\frac{u'+v}{1+vu'/c^2})

Now some simple bedmas and you're done...

 

[chris_0101]

This is what I got:

(u'^2 + 2u'v + v^2)/((u'^2 v^2)/c^4) + (2((u'v)/c^2) +1

Is this right?