Square root calculator
The cube root of −27 is −3, because −3 × −3 × −3 gives −27. Any odd root of a negative number is real. Even roots are not: no real number squared gives −4, which is why the square root of a negative belongs to the complex numbers. This tool says so explicitly rather than returning a bare NaN.
The square root of 144 is 12, because 12 × 12 = 144. The nth root of a number x is x raised to the power of 1/n. Odd roots of negative numbers are real. The cube root of −27 is −3, but even roots of negatives are not.
How to find a root
A perfect square is one whose root is a whole number, and the first few are worth recognising on sight: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144. Spotting them makes simplifying surds much faster. √72 becomes √36 × √2, which is 6√2. The tool flags whether a root came out whole for exactly that reason, since a non-whole result usually means the expression can be simplified rather than evaluated.
Questions
Twelve. It is a perfect square.
Not within the real numbers. The result is imaginary, √−4 is 2i.
That is real. The cube root of −27 is −3.
The number that, raised to the power n, gives your value. It equals x^(1/n).
Pull out the largest perfect square factor. √72 is √36 × √2, which is 6√2.