Can anyone give a complete proof step by step that the:
sqrt(41)+5=irrational
given that the sqrt(41) is irrational..
Show that it is irrational by the fact that the
sqrt(41) is irrational.
You have it right there. First, you are given that $\displaystyle \sqrt{41}$ is irrational. Then, you said that $\displaystyle \sqrt{41}=\frac{p}{q}-5$ is rational, since a rational subtract a rational is rational. Hence, $\displaystyle \sqrt{41}$ is rational and irrational.
Isn't this a problem?