  Let $E$ be an arbitrary directed graph and let $L$ be the Leavitt path
algebra of the graph $E$ over a field $K$. It is shown that every ideal of $L$
is an intersection of primitive/prime ideals in $L$ if and only if the graph
$E$ satisfies Condition (K). Uniqueness theorems in representing an ideal of
$L$ as an irredundant intersection and also as an irredundant product of
finitely many prime ideals are established. Leavitt path algebras containing
only finitely many prime ideals and those in which every ideal is prime are
described. Powers of a single ideal $I$ are considered and it is shown that the
intersection ${\displaystyle\bigcap\limits_{n=1}^{\infty}}I^{n}$ is the largest
graded ideal of $L$ contained in $I$. This leads to an analogue of Krull's
theorem for Leavitt path algebras.
