I have just seen the 2 remaining contestants on this show asked what the cube root of 512 is - given the choice of 8, 12 or 16 they could not work it out. These were not spotty teenagers, they looked middle-aged.
I despair for this country sometimes.
I'm inclined to agree but, since I left school nearly 50 years ago (shock horror), I cannot remember ever needing to work out a cube root . Other aspects of algebra and geometry yes, all the time. Maybe I've led a sheltered life never needing to know the volume of a sphere.
It's probably easier to see the answer to these things when you are relaxed at home than under time pressure with the camera and lights on you.
There's not really a need to do precise calculations on this one. If given the choice of three try the middle one first. 12 x 12 is 144 so multiplying by 12 aain is going to give something well over 1000, so 12 is too big. So the answer is 8. You don't really need to check that 8 x8 x8 is 512.
Or you can try an easier number. Try 10 x 10 x 10. That's 1000 which is too big. So answer is less than 10. Given the choice it must be 8.
I always encourage students (but they tend to ignore me) to do an estimate first using rounded numbers, so they can then check their final answer makes sense.
The big problem with this question though is most people don't know what a square root or cube root is. You can get GCSE maths without knowing it and the vast marity of us will never use square roots or cube roots again in our lives.