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
sign and xa
Roots are like exponents, only backward. For example,
to square the number 3 is to multiple 3 by itself two times: 32
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
Square roots are the most commonly used roots, but there
are also cube roots (numbers raised to 1
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
and fourth roots
. These roots of higher degrees operate
the same way square roots do. Because 33
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:
A number between 1 and 10.
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
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
Scientific Notation and Calculators
On many calculators, scientific notation is written differently
from what you’ve seen here. Instead of 3.1
your calculator might read 3.1 E33. The capital letter “E” has the same
role as the “
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.