One algorithm that is simple, but not very efficient, is to try out different numbers. Example, cubic root of 10:
2 cubed is 8, which is less than 10
3 cubed is 27, which is greater than 10
So the cubic root of 10 would be somewhere between 2 and 3; you can continue narrowing it down that way.
A calculator or computer program such as Excel would probably solve a cubic root - or any other root for that matter - using powers. For example, the cubic root of 10 is the same as 10 to the power (1/3).
Powers, in turn, are calculated using infinite series, using the formula:
a ^ b = exp(b ln a)
In this formula, exp() is the exponential function (e to the power ...), and "^" stands for power.
An "infinite series" is not as bad as it sounds; you don't actually calculate an infinite number of terms. Rather, the terms in an infinite series get smaller and smaller, once the terms are smaller than the desired accuracy, you can stop. It's actually a bit more complicated than that, but that's the basic idea.
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All numbers have cube roots (not necessarily integral cube roots) so every prime has cube roots.
Yes.
Many calculators will offer a way for you to do cube roots, or what number to the power of 3 will equal that number. There is no easy way to go about finding cube roots without a calculator, so I suggest not trying too hard.. but the cube root of 7 is 1.912931182772389 found from using a simple Cube Root calculator you can find on the internet.
4
The answer depends on "different from WHAT?" Positive cube roots, or negative square roots?