Find the closest integer to $$(3+\sqrt7)^4$$ by hand, without knowing the correct value of $\sqrt7$ (Maybe just knowing that $2<\sqrt7<3$).
My work:$$(3+\sqrt7)^4 = (16+6\sqrt7)^2 = 508+192\sqrt7$$
The "influence" of the uncertain $\sqrt7$ is pretty big if we just expand it out. Please help!
If you know that $2< \sqrt{7}<3$, then you know that $3-\sqrt{7}$ is less than one. So $(3-\sqrt{7})^4$ is even smaller. So if you add
$$(3+\sqrt{7})^4 + (3-\sqrt{7})^4$$
you don't change the number by very much, but there are no $\sqrt{7}$'s in the result.