What is the cubed root of 0.1?

This is a question you may be asked in an Oxbridge interview for a course with any mathematical aspect. They are expecting you to find an answer without a calculator, but more importantly than finding the answer, they are looking for how you reach the answer. Firstly try thinking of how else you could think of 0.1^1/3. One way you could think of this is 1^1/3/10^1/3. Since 1^1/3 is 1, all you need to do is figure out the cubed root of 10. Without a calculator you can't get an exact number, but that's okay. Think about the cubed numbers you do know, 1³ = 1, 2³ = 8, 3³ = 27. From this we can see that 10^1/3 must lie between 2 and 3, and much closer to 2 than 3. We could estimate that 10^1/3 is around 2.1. Now, all that's left to do is calculate 1^1/3/10^1/3 or 1/2.1. Again, we can't calculate 1/2.1 exactly, but we do know what something similar is - 1/2 = 0.5 and 1/3 = 0.333. So we know that 1/2.1 is less than 0.5, but not by a huge degree and is greater than 0.333. Again, we can estimate that 1/2.1 is close to 0.5 and is about 0.45. The true answer to 0.1^1/3 is 0.4641, so we're very close. In an Oxbridge interview, they don't expect you to know this off by heart or to be able to figure out the exact answer to 10 decimal places, they're much keener to see you talk through the process we just went to in order to see how you think.