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Integer Exponents

In order to indicate a product, centered dot, \( \cdot \), or parentheses around one or more symbols are used. Sometimes the symbol for multiplication is omitted. For instance, the product of $\mathbf{x}$ and $\mathbf{y}$ can be written in the following ways:

\(xy\)         \((x)y\)     \( x(y)\)      \((x)(y)\)    \(x \cdot y\)
The numbers $\mathbf{x}$ and $\mathbf{y}$ are called factors of the product $\mathbf{xy}$.
Suppose we have the product of two factors, each being $\mathbf{a}$. We can use the notation \(a^2\) to indicate this product, where the number 2 written at the upper right of the symbol $\mathbf{a}$ is called an exponent.
\(a^2 = a \cdot a\)          \(a^3 = (a)(a)(a)\)
Therefore, \(5^2 = 5 \cdot 5\); that is, \(5^2 = 25\).
In general, if $\mathbf{x}$ is a real number and $\mathbf{n}%$ is a positive integer,
            \( x^n = x \cdot x \cdot x \cdot \cdot \cdot x\)   ($\mathbf{n}$ factors of $\mathbf{x}$)
where $\mathbf{n}$ is called the exponent, $\mathbf{x}$ is called the base, and $\mathbf{x^n}$ is called the $\mbox{nth power of x}$. Hence, \(a^2\) is the second power of $\mathbf{a}$ and \(b^7\) is the seventh power of $\mathbf{b}$.
When a variable or symbol is written without an exponent, the exponent is understood to be 1. Hence, \(a = a^1\). Moreover, \(a^2\) is usually read as “a squared” and \(a^3\) is read as “a cubed.”
Historically, the representation of positive-integer powers by exponents was introduced by Rene Descartes (1596 - 1650) in 1637.
$Theorem$ $1A$
If $\mathbf{n}$ and $\mathbf{m}$ are positive integers and $\mathbf{a}$ is a real number, then
\[ (a^n)(a^m) = a^{n+m} \]

$Illustrative$ $Examples$
\( (3^4)(3^2)  = 3^{4+2} \)
           \( = 3^6\)
            \(= 729\)
\((x^4)(x)  = x^{4+1}\)
           \( = x^5\)
Perform the indicated operation. 
\((4a^4b^3)(-5a^5b^2) = [4(-5)][(a^4a^5)(b^3b^2)]\)
                                   \( = -20a^{4+5}b^{3+2}\)
                                    \(= -20a^9b^5\)

\((5nx^2n)(3nx^n)           = [(5)(3)][(nn)(x^2nx^n)]\)
                                   \( = 15n^{1+1}x^{2n+n}\)
                                    \(= 15n^2 x^3n\)

Suppose we have \((a^2)^3\). By the definition of a positive-integer exponent,
\((a^2)^3 = (a^2) (a^2) (a^2)\)
             \(= a^{2+2+2}\)
             \(= a^{(2)(3)}\)
             \(= a^6\)
Therefore, the product of $\mathbf{2}$ and $\mathbf{3}$ is the exponent of a after simplification. It is summarized in the following theorem.
$Theorem$ $1B$
If n and m are positive integers and a is a real number, then
(a^n)^m = a^[(n)(m)]
The following theorems definitely describe the behaviors of integer exponents. Applications and implications of these theorems are straightforward.
$Theorem$ $1C$
If n is a positive integer and a and b are real numbers, then
(ab)^n = a^nb^n