How totally radical!

$\large \begin{aligned} P & = \sqrt 2 \times \sqrt [4]4 \times \sqrt[4]{324} \times \sqrt[5]{7776} \times \sqrt[6]{256} \\ & = 2^\frac 12 \times (2^2)^\frac 13 \times (2^23^4)^\frac 14 \times (2^53^5)^\frac 15 \times (2^8)^\frac 16 \\ & = 2^{\frac 12+\frac 23+\frac 24+\frac 55+\frac 86}3^{\frac 44+\frac 55} \\ & = 2^43^2 \\ & = \boxed{144} \end{aligned}$

The first and third terms multiply to 6.

The second and fifth terms multiply to 4.

The fourth term is equal to 6.

6 * 4 * 6 = 144