In mathematics, we can think of radicals as the inverse operation to exponents.  If we have x², then we have x as a factor two times; if we take the square root of x² (or radical 2 of x²), we get x, the number when multiplied by itself gives us the number inside a radical sign (called the radicand).

Radicals take the form (a is real; n is a natural number >=2): 

• a is the radicand; it is the number we want to find the roots for
• n is which root we want to find 
• n can be any natural number >= 2 
• by default (that is, if there is no n specificed) the root is 2, which we call the "square root;" for example:   
• when n = 3, we are expressing a "cube root," that is, a number taken as a factor 3 times to give us the radicand; for example, the cube root of 8 = = 2 (since 2³ = 8)
• in the real number system, when n is even and the radicand is a real number, then a must be positive, since taking any number as a factor an even number of times gives us a positive number
• in the real number system, when n is odd and the radicand is a real number, then the radicand may be positive or negative (for example, we can find the cube root of -8, which is -2, since (-2)³ = -8). 

Note that when we use the radical sign, we want the primary root of the radicand, which is the positive root.  For example, if I have x² = 25, then x may be +5 or - 5.  However, if I write , then this expression evaluates to +5.  If we want -5, we would put a negative sign in front of the radical: -.

We can also write radicals as fractional exponents using this model: 

This says we take the nth root of a and raise it to power m.

For our purposes, let us look at the case when m is 1:

•  25^1/2 = = 5.
• 8^1/3 = = 2 

It is easy to find the square root of a number if it is a perfect square or the cube root of a number if it is a perfect cube.  However, if the radicand is not perfect, we can try to estimate the value.  We can also simplify radicals.

These class notes provide further explanation and examples.