Just as it's helpful to know the squares, it can also be very helpful to know some cubes. The cube root however, allows for negatives under the root unlike the square root. Without a calculator, we can estimate what the value could be and then we can start testing some numbers. Knowing 53=125 and 103=1000, 343 sits somewhere in between, closer to 125. By hand you can easily compute 6x6x6=216. So we can try 7x7x7 and we get 343. However, remember it was negative so we want -7. So the cube root of -343=-7.
Hope this helps!
