Let x = 1111111. We can then restate the original expression as

\[\sqrt{x\cdot 10^7+2x-3x}. \]

This then simplifies as

\[ \begin{eqnarray*} \sqrt{x \cdot 10^7+2x-3x} & = & \sqrt{x(10^7-1)}, &=& \sqrt{x \cdot 9999999}, \\ &=& \sqrt{x\cdot 9x}, \\ &=& \sqrt{9 \cdot x^2}, \\ &=& \sqrt{9 \cdot 1111111^2}, \\ &=& 3333333. \end{eqnarray*} \]