Elements of Algebra

62 ELEMENTS SECT. I. stance, $a$ is called the root, $b$ the exponent, and $a^b$ the power. 214. Farther, if we represent this power also by the letter $c$, we have $a^b = c$, an equation in which three letters $a$, $b$, $c$, are found; and we have shewn in treating of powers, how to find the power itself, that is, the letter $c$, when a root $a$ and its exponent $b$ are given. Suppose, for example, $a=5$, and $b=3$, so that $c = 5^3$: then it is evident that we must take the third power of 5, which is 125, so that in this case $c = 125$. 215. We have now seen how to determine the power $c$, by means of the root $a$ and the exponent $b$; but if we wish to reverse the question, we shall find that this may be done in two ways, and that there are two different cases to be considered: for if two of these three numbers $a$, $b$, $c$, were given, and it were required to find the third, we should immediately perceive that this question would admit of three different suppositions, and consequently of three solutions. We have considered the case in which $a$ and $b$ were the given numbers; we may therefore suppose farther that $c$ and $a$, or $c$ and $b$, are known, and that it is required to determine the third letter. But, before we proceed any farther, let us point out a very essential distinction between involution and the two operations which lead to it. When, in addition, we reversed the question, it could be done only in one way; it was a matter of indifference whether we took $c$ and $a$, or $c$ and $b$, for the given numbers, because we might indifferently write $a+b$, or $b+a$; and it was also the same with multiplication; we could at pleasure take the letters $a$ and $b$ for each other, the equation $ab=c$ being exactly the same as $ba=c$: but in the calculation of powers, the same thing does not take place, and we can by no means write $b^a$ instead of $a^b$; as a single example will be sufficient to illustrate: for let $a=5$, and $b=3$; then we shall have $a^b = 5^3 = 125$; but $b^a = 3^5 = 243$: which are two very different results. 216. It is evident then, that we may propose two questions more: one, to find the root $a$ by means of the given power $c$, and the exponent $b$; the other, to find the exponent $b$, supposing the power $c$ and the root $a$ to be known. 217. It may be said, indeed, that the former of these questions has been resolved in the chapter on the extraction of roots; since if $b=2$, for example, and $a^2=c$, we know by this means, that $a$ is a number whose square is equal to $c$, and consequently that $a = \sqrt{c}$. In the same manner, if G