find the distance from the origin to the point R(m^2-n^2 , 2mn) in simplest form
i got to m^2+2mn-n^2
but i think thats wrong
the answer is m^2 + n^2
I took red dog's work and expanded it for you. Credits go to him, it's just a few more steps so you can see what's going on.
$\displaystyle \sqrt{(m^2-n^2)^2+4m^2n^2}$
$\displaystyle =\sqrt{(m^2 - n^2)(m^2 - n^2) + 4m^2n^2}$
$\displaystyle =\sqrt{m^4 -2m^2n^2 + n^4 + 4m^2n^2}$
$\displaystyle =\sqrt{m^4+n^4+2m^2n^2}$
$\displaystyle =\sqrt{(m^2+n^2)^2}=m^2+n^2$
Well you can combine $\displaystyle -2m^2n^2$ with $\displaystyle 4m^2n^2$ since both terms have the same variables to the same power. There for you add the coefficients:
$\displaystyle (-2 + 4)m^2n^2 = 2m^2n^2$
From there we have:
$\displaystyle \sqrt{m^4 + n^4 + 2m^2n^2}$
All we are doing from here is factoring that into:
$\displaystyle \sqrt{(m^2 + n^2)^2}$
You can also use substitution if you can't see it right away. Ex: Let $\displaystyle m^2 = a$ $\displaystyle n^2 = b$
That gets you:
$\displaystyle \sqrt{a^2 + 2ab + b^2} = \sqrt{(a + b)^2}$ then resubstitute m and n back in: $\displaystyle \sqrt{(m^2 + n^2)^2}$
Either way, you come up with the same answer.