Part I. Section I. Chapter 16. “Of Powers in general.”

168 The product which we obtain by multiplying a number once, or several times by itself, is called a power. Thus, a square which arises from the multiplication of a number by itself, and a cube which we obtain by multiplying a number twice by itself, are powers. We say also in the former case, that the number is raised to the second degree, or to the second power; and in the latter, that the number is raised to the third degree, or to the third power.

169 We distinguish those powers from one another by the number of times that the given number has been multiplied by itself. For example, a square is called the second power, because a certain given number has been multiplied by itself; and if a number has been multiplied twice by itself we call the product the third power, which therefore means the same as the cube; also if we multiply a number three times by itself we obtain its fourth power, or what is commonly called the biquadrate: and thus it will be easy to understand what is meant by the fifth, sixth, seventh, etc. power of a number. I shall only add, that powers, after the fourth degree, cease to have any other but these numeral distinctions.

170 To illustrate this still better, we may observe, in the first place, that the powers of 1 remain always the same; because, whatever number of times we multiply 1 by itself, the product is found to be always 1. We shall therefore begin by representing the powers of 2 and of 3, which succeed each other as in the following order:

Powers. Of the number 2. Of the number 3.
1st 2 3
2nd 4 9
3rd 8 27
4th 16 81
5th 32 243
6th 64 729
7th 128 2187
8th 257 6561
9th 512 19683
10th 1024 59049
11th 2048 177147
12th 4096 531441
13th 8192 1594323
14th 16384 4782969
15th 32768 14348907
16th 65536 43046721
17th 131072 129140163
18th 262144 387420489

But the powers of the number 10 are the most remarkable: for on these powers the system of our arithmetic is founded. A few of them ranged in order, and beginning with the first power, are as follow:

1st 2nd 3rd 4th 5th 6th
10 100 1000 10000 100000 1000000 etc.

171 In order to illustrate this subject, and to consider it in a more general manner, we may observe, that the powers of any number, \(a\), succeed each other in the following order

1st 2nd 3rd 4th 5th 6th
\(a\) \(aa\) \(aaa\) \(aaaa\) \(aaaaa\) \(aaaaaa\) etc.

But we soon feel the inconvenience attending this manner of writing the powers, which consists in the necessity of repeating the same letter very often, to express high powers; and the reader also would have no less trouble, if he were obliged to count all the letters, to know what power is intended to be represented. The hundredth power, for example, could not be conveniently written in this manner; and it would be equally difficult to read it.

172 To avoid this inconvenience, a much more commodious method of expressing such powers has been devised, which, from its extensive use, deserves to be carefully explained. Thus, for example, to express the hundredth power, we simply write the number 100 above the quantity, whose hundredth power we would express, and a little towards the right-hand; thus \(a^{100}\) represents \(a\) raised to the 100th power, or the hundredth power of \(a\). It must be observed, also, that the name exponent is given to the number written above that whose power, or degree, it represents, which, in the present instance, is 100.

173 In the same manner, \(a^2\) signifies \(a\) raised to the 2nd power, or the second power of \(a\), which we represent sometimes also by \(aa\), because both these expressions are written and understood with equal facility; but to express the cube, or the third power \(aaa\), we write \(a^3\), according to the rule, that we may occupy less room; so \(a^4\) signifies the fourth, \(a^5\) the fifth, and \(a^6\) the sixth power of \(a\).

174 In a word, the different powers of a will be represented by

\[a, a^2, a^3, a^4, a^5, a^6, a^7, a^8, a^9, a^{10}, \textrm{etc.}\]

Hence we see that in this manner we might very properly have written \(a^1\) instead of \(a\) for the first term, to show the order of the series more clearly. In fact, \(a^1\) is no more than \(a\), as this unit shows that the letter \(a\) is to be written only once. Such a series of powers is called also a geometrical progression, because each term is by one-time, or term, greater than the preceding.

175 As in this series of powers each term is found by multiplying the preceding term by \(a\), which increases the exponent by 1; so when any term is given, we may also find the preceding term, if we divide by \(a\), because this diminishes the exponent by 1. This shows that the term which precedes the first term \(a^1\) must necessarily be \(\frac{a}{a}\), or 1; and, if we proceed according to the exponents, we immediately conclude, that the term which precedes the first must be \(a^0\); and hence we deduce this remarkable property, that \(a^0\) is always equal to 1, however great or small the value of the number \(a\) may be, and even when a is nothing; that is to say, \(a^0\) is equal to 1.

176 We may also continue our series of powers in a retrograde order, and that in two different ways; first, by dividing always by a; and secondly, by diminishing the exponent by unity: and it is evident that, whether we follow the one or the other, the terms are still perfectly equal. This decreasing series is represented in both forms in the following Table, which must be read backwards, or from right to left.

\(\frac{1}{aaaaaa}\) \(\frac{1}{aaaaa}\) \(\frac{1}{aaaa}\) \(\frac{1}{aaa}\) \(\frac{1}{aa}\) \(\frac{1}{a}\) \(1\) \(a\)
1st \(\frac{1}{a^6}\) \(\frac{1}{a^5}\) \(\frac{1}{a^4}\) \(\frac{1}{a^3}\) \(\frac{1}{a^2}\) \(\frac{1}{a^1}\)
2nd \(a^{-6}\) \(a^{-5}\) \(a^{-4}\) \(a^{-3}\) \(a^{-2}\) \(a^{-1}\) \(a^0\) \(a^1\)

177 We are now come to the knowledge of powers whose exponents are negative, and are enabled to assign the precise value of those powers. Thus, from what has been said, it appears that

\(a^0\) is equal to 1
\(a^{-1}\) is equal to \(\frac{1}{a}\)
\(a^{-2}\) is equal to \(\frac{1}{aa}\) or \(\frac{1}{a^2}\)
\(a^{-3}\) is equal to \(\frac{1}{a^3}\)
\(a^{-4}\) is equal to \(\frac{1}{a^4}\), etc.

178 It will also be easy, from the foregoing notation, to find the powers of a product, \(ab\); for they must evidently be

\(ab\), or \(a^1b^1\), \(a^2b^2\), \(a^3b^3\), \(a^4b^4\), \(a^5b^5\), etc.

and the powers of fractions will be found in the same manner; for example, those of \(\frac{a}{b}\) are

\(\frac{a^1}{b^1}\), \(\frac{a^2}{b^2}\), \(\frac{a^3}{b^3}\), \(\frac{a^4}{b^4}\), \(\frac{a^5}{b^5}\), \(\frac{a^6}{b^6}\), \(\frac{a^7}{b^7}\), etc.

179 Lastly, we have to consider the powers of negative numbers. Suppose the given number to be \(-a\); then its powers will form the following series:

\(-a\), \(+a^2\), \(-a^3\), \(+a^4\), \(-a^5\), \(+a^6\), etc.

Where we may observe, that those powers only become negative, whose exponents are odd numbers, and that, on the contrary, all the powers, which have an even number for the exponent, are positive. So that the third, fifth, seventh, ninth, etc. powers have all the sign -; and the second, fourth, sixth, eighth, etc. powers are affected by the sign +.


  1. Leonhard Euler. Elements of Algebra. Translated by Rev. John Hewlett. Third Edition. Longmans, Hurst, Rees, Orme, and Co. London. 1822.
  2. Leonhard Euler. Vollständige Anleitung zur Algebra. Mit den Zusätzen von Joseph Louis Lagrange. Herausgegeben von Heinrich Weber. B. G. Teubner. Leipzig and Berlin. 1911. Leonhardi Euleri Opera omnia. Series prima. Opera mathematica. Volumen primum.