5. Let $a>b>0, \sqrt[6]{a b}=4, \sqrt[6]{a}+\sqrt[6]{b}=2 \sqrt{7}$. If $[x]$ denotes the greatest integer less than or equal to $x$, then $[a]$ is ( ).
(A) 7037
(B) 7038
(C) 7039
(D) 7040
5.C.

According to the problem, $\sqrt[5]{a} 、 \sqrt[6]{b}$ are the two real roots of the equation $t^{2}-2 \sqrt{7} t+4=0$.
Solving this equation, we get
$$
a=(\sqrt{7}+\sqrt{3})^{6}, b=(\sqrt{7}-\sqrt{3})^{6} \text {. }
$$

Let $m=\sqrt{7}+\sqrt{3}, n=\sqrt{7}-\sqrt{3}$, then we have
$$
m+n=2 \sqrt{7}, m n=4 \text {. }
$$

Thus, $m^{6}+n^{6}=\left(m^{2}+n^{2}\right)\left(m^{4}-m^{2} n^{2}+n^{4}\right)$
$=\left[(m+n)^{2}-2 m n\right]\left[\left(m^{2}+n^{2}\right)^{2}-3 m^{2} n^{2}\right]$
$=(28-8)\left[(28-8)^{2}-3 \times 4^{2}\right]=7040$.
Since $0<n<1$, we have
$[a]=\left[m^{6}\right]=7039$.