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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