# 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