I'm wondering about the cube root trick:
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.
No 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.
How can you proof that this trick works?
If you only look at the last digit (modulo $10$), you have the pattern: $$0^3\equiv_{10}0$$ $$1^3\equiv_{10}1$$ $$2^3\equiv_{10}8$$ $$3^3\equiv_{10}7$$ $$4^3\equiv_{10}4$$ $$5^3\equiv_{10}5$$ $$6^3\equiv_{10}6$$ $$7^3\equiv_{10}3$$ $$8^3\equiv_{10}2$$ $$9^3\equiv_{10}1$$
For instance, $\sqrt[3]{140608}=52$, since $5^3<140<6^3$, and the last digit $8$ corresponds to the $2$ (see pattern). The reason why you don't have to think about the three last digits is because $$(a\cdot 10+b)^3=a^3\cdot 1000+3a^2b\cdot 100+3ab^2\cdot 10+b^3$$ So the first three digits is greater than $a^3$ (which means that this method would become hard to use for large numbers). This only works when you know that the number you're given is a perfect cube.