Page:Elementary algebra (1896).djvu/21

the coefficient of $bc$. A coefficient which is not merely numerical is sometimes called a literal coefficient.

Note. When the coefficient is unity it is usually omitted, and we write simply $a$, instead of $1a$.

7. A power of a quantity is the product obtained by repeating that quantity any number of times as a factor, and is expressed by writing the number of factors to the right of the quantity and above it. Thus,

$a\times a$ is called the second power of $a$, and is written $a^{2}$;

$a\times a\times a$ is called the third power of $a$, and is written $a^{3}$;

The index or exponent is the number which expresses the power of any quantity. Thus $2$, $5$, $7$ are respectively the indices of $a^{2}$, $a^{5}$, $a^{7}$.

Note. $a^{2}$ is usually read "$a$ squared"; $a^{3}$ is read "$a$ cubed"; $a^{3}$ is read "$a$ to the fourth"; and so on.

When the index is unity it is omitted, and we write simply $a$, instead of $a^{1}$. Thus $a$, $1a$, $a^{1}$, and $1a^{1}$ all have the same meaning.

8. The beginner must be careful to distinguish between coefficient and index.

Ex. 1. What is the difference in meaning between $3a$ and $a^{3}$?

By $3a$ we mean the product of the quantities $3$ and $a$,

By $a^{3}$ we mean the product of the quantities $a$, $a$, $a$.

Thus, if $a=4$,

${\begin{aligned}3a&=3\times a=3\times 4=12{\mbox{;}}\\a^{3}&=a\times a\times a=4\times 4\times 4=64\end{aligned}}$.

Ex. 2. If $b=5$, distinguish between $4b^{2}$ and $2b^{4}$.

Here⁠$4b^{2}=4\times b\times b=4\times 5\times 5=100$;
whereas $2b^{4}=2\times b\times b\times b\times b=2\times 5\times 5\times 5\times 5=1250$.

Ex. 3. If $x=1$, find the value of $5x^{4}$.

Here⁠$5x^{4}=5\times x\times x\times x\times x=5\times 1\times 1\times 1\times 1=5$.

Note. The beginner should observe that every power of 1 is 1.

Ex. 4. If $a=4$, $x=1$, find the value of $5x^{a}$.

$5x^{a}=5\times x^{a}=5\times 1^{4}=5\times 1=5$.