The rule
ⁿ√x = the number that gives x when raised to n
Why it works
A root of index n undoes raising to the power n. Which numbers have real roots depends on whether that index is odd or even: odd roots exist for every real number, while even roots need a non-negative starting value.
How to do it by hand
- Break the number into prime factors
- Any factor appearing n times comes out of the radical once
- What is left stays inside
- For a negative number with an odd index, the answer is negative
What is worth knowing
The odd–even distinction is the thing worth carrying away. The cube root of −8 is −2, because (−2)³ = −8, and that is a perfectly ordinary real number. The square root of −8 has no real value at all, because squaring always kills the sign. That is why odd-degree polynomials always have at least one real root and even-degree ones need not. In practice, roots of index 4 or higher are usually computed as fractional exponents, since x^(1/n) is the same thing and computers handle it directly.