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:
sign as the radical
as the radicand.
Roots are like exponents, only backward. For
example, to square the number 3 is to multiply 3 by itself:
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
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
, fourth roots
, 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).