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 x^a 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 is a convention used to express large numbers. A number written in scientific notation has two parts:

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.

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.