is this question Using chinese remainder theorem? 
I think that it will use Chinese remainder theorem
but I don't know how to put...
well by CRT, there exist $x=k\mod (m_1)(m_2)(m_3)$
which is $x=a_1^3 \mod m_1$ and $x=a_2^3 \mod m_2$ and $x=a_3^3 \mod m_3$
 but $k$ must be $a^3$ ?

Yes, one uses the Chinese Remainder Theorem, but not quite in the way partly described in the post.
Consider the system of congruences $y\equiv a_i\pmod{m_i}$ ($i=1,2,3$). By the Chinese Remainder Theorem, this system of congruences has a solution. Call it $a$. 
Then $a^3\equiv a_i^3\equiv x\pmod{m_i}$ for all $i$, and therefore $a^3\equiv x\pmod{m_1m_2m_3}$.
Remark: The above answers the question as given. However, the wording of the question seems a little peculiar. It says "Suppose that there exists $x$ satisfying $\dots$." However, given $a_1,a_2,a_3$, there always exists such an $x$ (Chinese Remainder Theorem), so it is odd to suppose that there is an $x$. 
