To find the closest integer to the cube root of a sum of two cubes, I can try to estimate the sum and see if it is close to a perfect cube. I know that $6^3=216$ and $8^3=512$, so $6^3+8^3=728$. The closest perfect cube to 728 that I can think of is $9^3=729$. If I take the cube root of both sides, I get $\sqrt[3]{728} \approx \sqrt[3]{729}=9$. Therefore, the integer that is closest to $\sqrt[3]{6^3+8^3}$ is 9. 
What integer is closest to the value of $\sqrt[3]{6^3+8^3}$?