FACTORS. COEFFICIENTS. POWERS.

Thus, $a^{1}$ , or simply $a$ , denotes that $a$ is taken once as a factor; $a^{2}$ denotes that $a$ is taken twice as a factor; $a^{3}$ denotes that $a$ is taken three times as a factor; and $a^{4}$ denotes that $a$ is taken four times as a factor; and so on. These are read: the first power of $a$ ; the second power of $a$ ; the third power of $a$ ; the fourth power of $a$ ; and so on.

$a^{3}$ is written instead of $a a a$ .

$a^{4}$ is written instead of $a a a a$ .

The meaning of coefficient and exponent must be carefully distinguished. Thus,

4a = a + a + a + a; a^4 = a× a× a× a. [1] lint If $a = 3$ , 4a = 3 + 3 + 3 + 3 = 12. a^4 = 3 × 3 × 3 × 3 = 81.

The second power of a number is generally called the square of that number; thus, $a^{2}$ is called the square of $a$ , because if $a$ denotes the number of units of length in the side of a square, $a^{2}$ denotes the number of units of surface in the square. The third power of a number is generally called the cube of that number; thus, $a^{3}$ is called the cube of $a$ , because if $a$ denotes the number of units of length in the edge of a cube, $a^{3}$ denotes the number of units of volume in the cube.