Roots and Radicals

　　We just saw that roots express fractional exponents. But it is often easier to work with roots in a different format. When a number or term is raised to a fractional power, the expression can be converted into one involving a root in the following way:

　　with the　sign as the radical sign, and　as the radicand.　　Roots are like exponents, only backward. For example, to square the number 3 is to multiply 3 by itself: 32 = 3　3 = 9. The root of 9, , is 3. In other words, the square root of a number is the number that, when squared, is equal to the given number. 　　Square roots are the most commonly used roots, but there are also cube roots (numbers raised to 1 3), fourth roots, fifth roots, etc. Each root is represented by a radical sign with the appropriate number next to it (a radical without any superscript denotes a square root). For example, cube roots are shown as , fourth roots as , and so on. These roots of higher degrees operate the same way square roots do. Because 33 = 27, it follows that the cube root of 27 is 3.Here are a few examples:

　　The same rules that apply to multiplying and dividing exponential terms with the same exponent apply to roots as well. Look for yourself:　　Just be sure that the roots are of the same degree (i.e., you are multiplying or dividing all square roots or all roots of the fifth power).