3.69: Binomial independent term and surd identity

Find and simplify the term independent of $x$ in the expansion of $\left(x^{2}-\dfrac{1}{2x^{3}}\right)^{10}$.

DO NOT USE A CALCULATOR. Use the binomial theorem to show that $(\sqrt{2}+1)^{4}+(\sqrt{2}-1)^{4}=k$, where $k$ is an integer to be found.

Hence write $\dfrac{1}{(\sqrt{2}+1)^{4}}+\dfrac{1}{(\sqrt{2}-1)^{4}}$ in the form $a+b\sqrt{2}$, where $a$ and $b$ are integers.