When to square a sum and when not to

[mess1n]

Hey, I've got a question which might be really simple, I'm not too sure yet!

Basically, I'm going over the compton scattering calculations, and there's a part where:

v - v' + (m_ec²)/h = $$\sqrt{something else}$$

Basically, the next step is to square both sides of the equation.

To do this, my lecturer squares the LHS as a sum (i.e. in the form (a+b)² instead of doing a² + b²... where a = (v - v') and b = (m_ec²)/h).

My question is... why do you in some instances take the square of the sum, and in other instances take the square of the individual components. I'm assuming there is a non-arbitary reason for this.. but I don't know about it!

Any help or pointers in the right direction would be much appreciated.

Cheers,
Andrew

 

[tiny-tim]

Welcome to PF!

Hey Andrew! Welcome to PF!

(have a square-root: √ )

My question is... why do you in some instances take the square of the sum, and in other instances take the square of the individual components. I'm assuming there is a non-arbitary reason for this.. but I don't know about it!

You always take the square of the sum.

(Though there are a few cases where that is the same as taking the sum of the squares, for example if they are perpendicular components of vectors )

 

[mess1n]

Cheers for the welcome, and for the answer!

 

[Nick89]

If you got
$$x = \sqrt{y}$$
and you square it to
$$x^2 = y$$

If x happens to be a sum
$$x = a + b$$
then you get
$$(a + b)^2 = y$$
and not
$$a^2 + b^2 = y$$

The reason is simple. In general,
$$a^2 + b^2 \neq (a + b)^2$$
because there's also the crossterm:
$$(a + b)^2 = a^2 + b^2 + 2ab$$