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 Logarithms
4.11 Review Questions
4.12 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 xa as the radicand.
Roots are like exponents, only backward. For example, to square the number 3 is to multiple 3 by itself two times: 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, and so on. 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 and fourth roots as . 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. Consider these examples:
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).
Scientific Notation
Scientific notation is a convention used to express large numbers. A number written in scientific notation has two parts:
  1. A number between 1 and 10.
  2. The power of 10 by which you must multiply the first number in order to obtain the large number that is being represented.
The following examples express numbers in scientific notation:
Scientific notation is particularly useful when a large number contains many zeros or needs to be approximated because of its unwieldy size. Approximating quantities in scientific notation can prevent unnecessarily messy calculations. Look at the following expression:
Finding the product would be pretty nasty—even if you’re using a calculator. Approximating each number using scientific notation makes the problem a lot easier:
When we compare this approximation to the actual product, we find that we were less than 1% off. Not too shabby.
Also, note the way in which we combined the terms in the last example to make the multiplication a little simpler:
In general terms:
Often, this type of simplification can make your calculations easier.
Scientific Notation and Calculators
On many calculators, scientific notation is written differently from what you’ve seen here. Instead of 3.1 1033, your calculator might read 3.1 E33. The capital letter “E” has the same role as the “ 10(power),” only it’s a little shorter. In general, scientific notation allows you to work with numbers that might either be tedious to manipulate or too large to fit on your calculator.
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