FACTORS. COEFFICIENTS. POWERS.

17. Factors.

When a number consists of the product of two or more numbers, each of these numbers is called a factor of the product.

The sign $\times$ is generally omitted between a figure and a letter, or between letters; thus, instead of $63\times a\times b$, we write $63ab$; instead of $a\times b\times c$, we write $abc$.

The expression $abc$ must not be confounded with $a+b+c$. $abc$ is a product; $a+b+c$ is a sum.

When a sign of operation is omitted in the notation of Arithmetic, it is always the sign of addition; but when a sign of operation is omitted in the notation of Algebra, it is always the sign of multiplication. Thus, $456$ means $400+50+6$, but $4ab$ means $4\times a\times b$.

Factors expressed by letters are called literal factors; factors expressed by figures are called numerical factors.

If one factor of a product is equal to $0$, the product is equal to $0$, whatever the values of the other factors. Such a factor is called a zero factor.

20. Coefficients.

A known factor of a product which is prefixed to another factor, to show the number of times that factor is taken, is called a coefficient.

Thus, in $7c$, $7$ is the coefficient of $c$; in $7ax$, $7$ is the coefficient of $ax$, or, if $a$ is known, $7a$ is the coefficient of $x$.

By coefficient, we generally mean the numerical coefficient with its sign. If no numerical coefficient is written, $1$ is understood. Thus, $ax$ means the same as $1ax$.

21. Powers and Roots.

A product consisting of two or more equal factors is called a power of that factor, and one of the equal factors is called a root of the number.

Thus, $9=3\times 3$; that is, $9$ is a power of $3$, and $3$ is a root of $9$.

22. Indices or Exponents.

An index or exponent is a number-symbol written at the right of, and a little above, a number.

If the index is a whole number, it shows the number of times the given number is taken as a factor.