To find a cube root of a large number, you only have to memorize the cube roots of numbers 1 to 10.
Take the last digit of your number, it'll be the last digit of the result.
Then ignore the last 3 digits, look at what remains, and find which of the first 10 cubes is the closest to it without going over.
Now put the result together.
Example: $\sqrt[3]{39304} = 34$ because last digit of 39304 is 4 and $3^3 = 27$ is closest cube to 39.
Why can we ignore the last 3 digits?
