Power is powerful

Sum of first $n$ square numbers $=\dfrac{n(n+1)(2n+1)}{6}$

$\implies \dfrac{10(11)(21)}{6(770)}=\boxed{\dfrac{1}{2}}$

$1^2 + 2^2 + 3^2 + 4^2 + 5^2 + 6^2 + 7^2 + 8^2 + 9^2 + 10^2 = 385$

$\frac{385}{770}$ $=$ $\frac{1}{2}$

Answer $\rightarrow$ $\frac{1}{2}$

Sum of first $n$ square numbers is:

$\dfrac{n(n+1)(2n+1)}{6}$

Then we have

\[\begin{align} \text{Expression} &= \dfrac{10(11)(21)}{6} \div 770 \\ &= \dfrac{385}{770} \\& = \boxed{\dfrac{1}{2}}

\end{align}\]