Mastering Algebra - Working With Exponents - Part I

Mastering algebra requires that the student be cognizant of the properties of exponents. Exponents occur repeatedly in algebra and indeed in all higher branches of mathematics. Here in this series of articles we discuss what an exponent is and how to handle and simplify expressions involving powers.

An exponent is the power of a number or expression. For example 3^4 (in which the "^" caret symbol represents exponentiation, or the raising to a power), the number 3 serves as the base, and the 4 after that special caret symbol tells us how many times to use 3 as a factor when multiplying by itself. Thus 3^4 means 3x3x3x3 = 81. Thus the exponent serves as a convenient shorthand notation to indicate repeated multiplication using the same number as multiplicand.

It is very easy to simplify expressions involving exponents, whether these be purely numerical examples as in (3^4)(3^2), or algebraic examples such as( x^3)(x^4). When the base is the same and we are multiplying expressions involving exponents, we simply add the exponents and retain the base. Thus in (3^4)(3^2), we do 4+2 = 6 and thus this expression becomes 3^6. In ( x^3)(x^4) we have 3+4 = 7 and thus this expression becomes x^7. If it is not obvious why we would add exponents together in such expressions, just think of the exponent as signifying beads on a necklace. If you string together 3 beads and then 4 beads, as in the second expression above, you have 7 beads.

If you have an expression in which you raise an exponential expression to another power, you simply multiply the exponents of the expression. Thus in (x^4)^2, you multiply the 4 and 2 to get 8, and end up with x^8. To understand why this is so, you need to recall that the exponent 2 in this example applied to the x^4 expression, tells us to use that twice to multiply itself. Multiplying x^4 by itself gives us x^8, as now we can use the rule learned in the previous paragraph. If you break things down this way and understand not only how but why, you are then in a much better position to make serious progress in algebra.

Two other key properties of exponents that you need to know are the following: 1) When you raise anything to the first power you obtain the given quantity; thus 3^1 = 3 and x^1 = x. This 1-exponent is also an invisible demon in the sense that even though we do not generally write the "1-exponent" it is always there understood. This is important to understand in examples such as x(x^5), which is really (x^1)(x^5) and thus equals x^6;2) Any expression to the 0th power is equal to 1. Thus x^0 = 1, and 4^0 = 1.

In the next part of this article, we shall explore the distributive and division properties of exponents. Once you get all the properties down pat, you will never again be at a loss with exponents; and since you will invariably come across exponents in all aspects of mathematics, having a mastery of this aspect will insure your continued success in this discipline.