Roots and Radicals
4.1 Order of Operations
4.2 Numbers
4.3 Factors
4.4 Multiples
4.5 Fractions
4.6 Decimals
4.7 Percents
4.8 Exponents
4.9 Roots and Radicals
4.10 Scientific Notation
4.11 Logarithms
4.12 Review Questions
4.13 Explanations
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 13), 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).
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