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.

177. We are thus brought to understand the nature of powers, whose exponents are negative, and are enabled to assign the precise value of these powers. From what has been said, it appears that,

178. It will be easy, from the foregoing notation, to find the powers of a product, a b. They must evidently be ab, or a1 b1, a2 b2, a3 b3, a1 b*, a' b3, &c. And the powers of fractions will

a

be found in the same manner; for example those of are,

b

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

—a,+aa, — a3, +a“, — a3, +a®, &c.

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, &c., powers have each the sign;-; and the second, fourth, sixth, eighth, &c. powers are affected with the sign +.

CHAPTER XVIII

Of the calculation of Powers.

180. We have nothing in particular to observe with regard to the addition and subtraction of powers; for we only represent these operations by means of the signs + and —, when the powers are different. For example, a3 +a3 is the sum of the second and third powers of a; and a3 -a is what remains when we subtract the fourth power of a from the fifth; and neither of these results can be abridged. When we have powers of the same kind, or degree, it is evidently unnecessary to connect them by signs; a3 + a3 makes 2 a3, &c.

181. But, in the multiplication of powers, several things require attention.

First, when it is required to multiply any power of a by a, we obtain the succeeding power, that is to say, the power whose exponent is greater by one unit. Thus a3, multiplied by a, produces a3; and a3, multiplied by a, produces a1. And, in the same manner, when it is required to multiply by a the powers of that number which have negative exponents, we must add 1 to the exponent. Thus, a ̄1 multiplied by a produces a° or 1; which is made more evident by considering that a1 is equal

to

1

a

1

1 and that the product of by a being, it is consequently equal to 1. Likewise a

1

multiplied by a produces a1, or

; and a 10, multipled by a, gives a, and so on.

182. Next, if it be required to multiply a power of a by a a, or the second power, I say that the exponent becomes greater by 2. Thus, the product of a2 by a2 is a; that of a2 by a3.

a; that of a by a is a; and, more generally, a multiplied by a2 makes an+2. With regard to negative exponents, we shall have a1, or a, for the product of a-1 by a'; for a-1 being equal

1

to, it is the same as if we had divided a a by a; consequently

a a

the product required is, or a. So a-3, multiplied by a pro

a

duces a°, or 1 ; and a-3, multiplied by a3, produces a-1.

183. It is no less evident that, to multiply any power of a by a3, we must increase its exponent by three units; and that consequently the product of a" by a3 is an+3. And whenever it is required to multiply together two powers of a, the product will be also a power of a, and a power whose exponent will be the sum of the exponents of the two given powers. For example, a multiplied by as will make ao, and a12 multiplied by a will produce a19, &c.

184. From these considerations we may easily determine the highest powers. To find, for instance, the twenty-fourth power of 2, I multiply the twelfth power by the twelfth power, because 224 is equal to 212 x 212. Now we have already seen that 212 is 4096; I say therefore that the number 16777216, or the product of 4096 by 4096, expresses the power required, 224.

185. Let us proceed to division. We shall remark in the first place, that to divide a power of a by a, we must subtract 1 from the exponent, or diminish it by unity. Thus a', divided by a, gives a1; ao, or 1, divided by a, is equal to a-1 or

divided by a, gives a ̄1.

1

; a-3,

186. If we have to divide a given power of a by a3, we must diminish the exponent by 2; and if by a3, we must subtract three units from the exponent of the power proposed. So, in general, whatever power of a it is required to divide by another power of a, the rule is always to ubtract the exponent of the second from the exponent of the first of these powers. Thus a1, divided by a', will give a ; a divided by a7, will give a-1; and a3, divided by a, will give a~7.

187. From what has been said above, it is easy to understand the method of finding the powers of powers, this being done by multiplication. When we seek, for example, the square, or the second power of a3, we find ao; and in the same manner we

find a1 for the third power or the cube of a*. To obtain the square of a power, we have only to double its exponent; for its cube, we must triple the exponent; and so on. The square of a" is a2n; the cube of a" is as"; the seventh power of a" is a7", &c.

188. The square of a3, or the square of the square of a, being a1, we see why the fourth power is called the bi-quadrate. The square of a3 is a ; the sixth power has therefore received the name of the square-cubed.

Lastly, the cube of a3 being ao, we call the ninth power the cubo-cube. No other denominations of this kind have been introduced for powers, and indeed the two last are very little used.

CHAPTER XVIII.

Of Roots with relation to Powers in general.

189. SINCE the square root of a given number is a number, whose square is equal to that given number; and since the cube root of a given number is a number, whose cube is equal to that given number; it follows that any number whatever being given, we may always indicate such roots of it, that their fourth, or their fifth, or any other power, may be equal to the given number. To distinguish these different kinds of roots better, we shall call the square root the second root; and the cube root the third root; because, according to this denomination, we may call the fourth root, that whose biquadrate is equal to a given number; and the fifth root, that whose fifth power is equal to a given number, &c.

3

190. As the square, or second root, is marked by the sign ✓, and the cubic or third root by the sign, so the fourth root is represented by the sign; the fifth root by the sign; and so on; it is evident that according to this mode of expression,

4

2

5

the sign of the square root ought to be. But as of all roots this occurs most frequently, it has been agreed, for the sake of brevity, to omit the number 2 in the sign of this root. So that

when a radical sign has no number prefixed, this always shews that the square root is to be understood.

191. To explain this matter still further, we shall here exhibit the different roots of the number a, with their respective values:

192. Whether the number a therefore be great or small, we know what value to affix to all these roots of different degrees.

It must be remarked also, that if we substitute unity for a, all those roots remain constantly 1; because all the powers of 1 have unity for their value. If the number a be greater than 1, all its roots will also exceed unity. Lastly, if that number be less than 1, all its roots will also be less than unity.

193. When the number a is positive, we know from what was before said of the square and cube roots, that all the other roots may also be determined, and will be real and possible numbers.

But if the number a is negative, its second, fourth, sixth, and all the even roots, become impossible, or imaginary numbers; because all the even powers, whether of positive, or of negative numbers, are affected with the sign +. Whereas the third, fifth, seventh, and all odd roots, become negative, but rational; because the odd powers of negative numbers, are also negative.