Past exam of the mathematics course of the University of Cambridge / 2015 / ii / Paper 2 / 21F / iv / Solution

The equality is false. In $A=k[x,y]$, take $I=(y)$ and $J=(y-x^2)$. Both are prime ideals, since their quotient rings are isomorphic to $k[x]$, so both are radical. ButThe element $x$ belongs to the latter ideal and not the former. This also shows why taking the radical of an ideal can enlarge a sum of already radical ideals.