0.3 Exponents  (Page 59/4)

Exponents

Introduction

Exponentiation is a mathematical operation that is employed extensively in applications in fields that range from science and engineering to finance and economics. This module will begin with a brief discussion of the terminology and the mathematics involved with exponentiation. This will be followed with some applications of exponentiation in science and engineering.

Scientific notation and engineering notation are introduced in this module as two common means for expressing physical quantities. Each of these representational schemes involve the use of exponents. Applications are presented on topics including electrical power, gravitatitational force and electrostatic force are presented as a means for illustrating how exponentiation can be useful in problem solving.

Definition and terminology

Exponentiation is an operation in mathematics that makes use of two numbers known as the base ( x ) and the exponent ( n ) and is expressed as

In this module, we will restrict our discussion to situations where the exponent ( n ) is an integer. Being an integer, the exponent may be positive, negative or equal to 0. We will consider each case below.

If the exponent n is a positive integer, then the operation of exponentiation is equivalent to the multiplication of the base x with itself a total of n times.

In the situation where the base ( x ) is 5 and the exponent ( n ) is 3, we obtain the result

Now, let us the case where the exponent is a negative integer. In this case, the process of exponentiation is equivalent to dividing one by the base x by n times.

Let us consider an example where the base ( x ) is 5, but now the exponent ( n ) is -3. In this example, we obtain the result

Suppose that we form the product of $5^3$ with $5^{-3}$ .

The result of the multiplication reminds us of the property of exponents that states whenever a base ( x ) is raised to an exponent ( n ) that is zero, the result is 1.

Scientific and engineering notation

One of the most important uses of exponents in the fields of science and engineering is that of scientific notation. As is often the case, in the fields of science and engineering one often deals with numbers that are extremely large or extremely small. Scientific notation is an effective means for writing such number that makes use of exponents. In many cases, scientific notation enables one to write very large or very small numbers in a manner that is more convenient, insightful and compact that writing numbers using decimal notation.

Numbers can be written in scientific notation in the following form

where a is the coefficient which can be any real number in the range 1<|a|<10 and b is an integer that represents the exponent.

Example (Avogadro’s Number)

In chemistry, the quantity of an element having a weight in grams numerically equal to that element’s atomic weight is the gram atomic weight of that element. This quantity is often refererred to as a gram atom . A gram atom of any element contains the same number of atoms as the gram atom of any other element. The number of atoms in any gram atom is called Avogadro’s number (N). Through meticulous experimental study, the value of Avogadro’s number has been determined as