7.1 Basic properties of exponents

$\begin{array}{ll}18\cdot 18\cdot 18\cdot 18\hfill & \text{Repeated}\text{multiplication}\text{of}18.\text{All}\text{four}\text{factors},18,\text{are}\text{the}\text{same}\text{.}\hfill \\ x\cdot x\cdot x\cdot x\cdot x\hfill & \text{Repeated}\text{multiplication}\text{of}x.\text{All}\text{five}\text{factors},x,\text{are}\text{the}\text{same}\text{.}\hfill \\ 3\cdot 7\cdot a\hfill & \text{Nonrepeated}\text{multiplication}\text{.}\text{None}\text{of}\text{the}\text{factors}\text{are}\text{the}\text{same}\text{.}\hfill \end{array}$

$7\cdot 7\cdot 7\cdot 7\cdot 7\cdot 7=7^6$ .

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.

$2yyy=2y^3$ .

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.

$8x^3$ 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.