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In Section [link] we were reminded that multiplication is a description for repeated addition. A natural question is “Is there a description for repeated multiplication?” The answer is yes. The notation that describes repeated multiplication is exponential notation .
$\begin{array}{ll}18\cdot 18\cdot 18\cdot 18\hfill & \text{Repeated}\text{\hspace{0.17em}}\text{multiplication}\text{\hspace{0.17em}}\text{of}\text{\hspace{0.17em}}18.\text{\hspace{0.17em}}\text{All}\text{\hspace{0.17em}}\text{four}\text{\hspace{0.17em}}\text{factors},\text{\hspace{0.17em}}18,\text{\hspace{0.17em}}\text{are}\text{\hspace{0.17em}}\text{the}\text{\hspace{0.17em}}\text{same}\text{.}\hfill \\ x\cdot x\cdot x\cdot x\cdot x\hfill & \text{Repeated}\text{\hspace{0.17em}}\text{multiplication}\text{\hspace{0.17em}}\text{of}\text{\hspace{0.17em}}x.\text{\hspace{0.17em}}\text{All}\text{\hspace{0.17em}}\text{five}\text{\hspace{0.17em}}\text{factors},\text{\hspace{0.17em}}x,\text{are}\text{\hspace{0.17em}}\text{the}\text{\hspace{0.17em}}\text{same}\text{.}\hfill \\ 3\cdot 7\cdot a\hfill & \text{Nonrepeated}\text{\hspace{0.17em}}\text{multiplication}\text{.}\text{\hspace{0.17em}}\text{None}\text{\hspace{0.17em}}\text{of}\text{\hspace{0.17em}}\text{the}\text{\hspace{0.17em}}\text{factors}\text{\hspace{0.17em}}\text{are}\text{\hspace{0.17em}}\text{the}\text{\hspace{0.17em}}\text{same}\text{.}\hfill \end{array}$
Exponential notation is used to show repeated multiplication of the same factor. The notation consists of using a superscript on the factor that is repeated . The superscript is called an exponent .
Note that the definition for exponential notation only has meaning for natural number exponents. We will extend this notation to include other numbers as exponents later.
$7\cdot 7\cdot 7\cdot 7\cdot 7\cdot 7={7}^{6}$ .
The repeated factor is 7. The exponent 6 records the fact that 7 appears 6 times in the multiplication.
$x\cdot x\cdot x\cdot x={x}^{4}$ .
The repeated factor is $x$ . The exponent 4 records the fact that $x$ appears 4 times in the multiplication.
$(2y)(2y)(2y)={(2y)}^{3}$ .
The repeated factor is $2y$ . The exponent 3 records the fact that the factor $2y$ appears 3 times in the multiplication.
$2yyy=2{y}^{3}$ .
The repeated factor is $y$ . The exponent 3 records the fact that the factor $y$ appears 3 times in the multiplication.
$(a+b)(a+b)(a-b)(a-b)(a-b)={(a+b)}^{2}{(a-b)}^{3}$ .
The repeated factors are $(a+b)$ and $(a-b)$ , $(a+b)$ appearing 2 times and $(a-b)$ appearing 3 times.
Write each of the following using exponents.
$a\cdot a\cdot a\cdot a$
${a}^{4}$
$(3b)(3b)(5c)(5c)(5c)(5c)$
${(3b)}^{2}{(5c)}^{4}$
$2\cdot 2\cdot 7\cdot 7\cdot 7\cdot (a-4)(a-4)$
${2}^{2}\cdot {7}^{3}{(a-4)}^{2}$
$8xxxyzzzzz$
$8{x}^{3}y{z}^{5}$
$8{x}^{3}$ means $8\cdot xxx$ and not $8x8x8x$ . The exponent 3 applies only to the factor $x$ since it is only to the factor $x$ that the 3 is connected.
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