231.

CHAP. XXVI.

OF THE CALCULATION OF POWERS.

THERE is nothing particular to be observed with regard to the addition and subtraction of powers; for those operations can only be represented by the signs + and — when the powers are different. For example, the sum of the second and third powers of a can only be represented by a3 +a2; and in like manner a3 —at is the only manner in which we can signify the difference between the fifth and fourth powers of a; neither of which results can be in any way abridged. Powers, however, of the same degree need not be connected by signs, since a+a3 is evidently 2a, and 5b3b is 2b, &c.

232. In the multiplication and division of powers, however, there are several circumstances that require particular attention.

First, as we have already seen in the preceding Chapter, when it is required to multiply any power of a by a, we obtain the next succeeding power, or that power whose index is increased by unity. Thus a2 multiplied by a produces a3 ; and a multiplied by a produces a*. In like manner also, if it be required to multiply by a any power of a whose index is negative, we have only to add one to that index, in order to obtain the next succeeding power. Thus a1 multiplied by a produces ao, which has been shown to be equal to 1, and which is thus made more apparent by the consideration that a1 is equivalent to, and that multiplied by a pro

a

1

a

duces 1. So also a multiplied by a will produce a―1,

1

a

ΟΙ and a-10 multiplied by a gives a-, and so on.

a

Next, if it be required to multiply any power of a by a2, it is equally clear that the index or exponent of that power must be increased by 2. Thus the product of a2 by a2 is a*; a3 by a2 is a3, &c. And the same rule will apply to the powers whose exponents are negative, viz. a-1 by a will give a; a~2

1

by a2 will give a°, or 1, and a→ by a2 will produce a−1 or α

233. It is no less evident, that to multiply any power of a by a3, we must increase the exponent of the power by 3, and that consequently the product of a3 by a3 is ao. And generally that a" multiplied by am will produce an+m. From which we deduce as a general rule, that the multiplication of the powers of A is effected by adding together their indices or exponents.

234. From these considerations it will be easily understood that, è converso, to divide any power of a by a, it is only necessary to subtract 1 from the index of the power; and that a2 divided by a is a2-1, or a; a3 divided by a is a3-1, or a2. Again, that to divide any power of a by a2 it will be requisite only to subtract 2 from the index, and generally that the quotient of a" divided by am is a"-m; the same being equally true with respect to the powers whose exponents are negative, and a3 divided by a is at, and a" divided by am is a−(n + m), From which is deduced also the general rule, that the division of powers is effected by the subtraction of their indices.

CHAP. XXVII.

OF ROOTS WITH RELATION TO POWERS IN GENERAL.

235. SINCE it has already been shown (Chap. XIV.) that the square root of a given number is a number such that its square is equal to that given number, and that the cube root of a given

number

number is a number such that its cube is equal to that given number, it follows that any number being given, we may always represent such roots of them that their fourth, fifth, or 7th powers may in like manner be equal to the number so given. In order therefore the better to distinguish these different kinds of roots, it is usual to term the square root, the second root, and the cube root the third root; since according to this denomination we may term the fourth root of a given number, that of which the fourth power is equal to the number so given, and so on of the fifth, sixth, seventh, and nth roots.

236. In the same manner therefore as the square or second root of a is signified by the sign a or a, and the third root by /a, the fourth root will be expressed by the sign /a, the fifth root bya, and generally the nth root by the sign /a.

Whether therefore the value of a be great or small, we can easily understand what value is to be attached to its different roots. And it will be observed, that if a be equal to unity or 1, its roots, as well as its powers, will be always 1; but that if it be greater than unity, its roots of whatever degree will always have a value greater than 1, and that if it be less than 1, all its roots will on the other hand be less than 1.

237. Now if a be positive, we have already seen that all its roots will be real and possible numbers; but on the contrary, if a be negative, then that the roots will be possible or impossible, according as the indices of the roots are odd or even. For instance, -a, -a, -a, &c. will be impossible quantities, because the square, the fourth power, &c. of negative quantities are always positive; but that 3/—a, -a, -a, &c. will, although negative, be yet possible and rational quantities, since the cube, fifth, and seventh powers of negative numbers are also negative.

238. We have here therefore an inexhaustible source of a new species of surds or irrational quantities, since whenever

the

the number a is not in reality such a power as any one of the foregoing indices would represent or seem to require, as in the case of -a, &c., it is then impossible to express that root by any numbers, either integral or fractional, and consequently every such root must be classed with those numbers which are denominated irrational or impossible.

CHAP. XXVIII.

ON THE METHOD OF REPRESENTING IRRATIONAL NUMBERS

239.

BY FRACTIONAL EXPONENTS.

WE have seen in the preceding Chapter that the square of any power is found by doubling its index or exponent; and that in general the square or second power of a" is an. The converse of this is therefore equally true; viz. that the square root of a2" is a"; or in general, that the square root of a power is found by taking the half of the index of that power, or in other words by dividing the index by 2.

n

Thus the square root of a2 is a' or a; of a1 is ao, of a¤ is a3, and so on; and since this is a general truth, we see that the square root of a3 must necessarily be a, and that of a, a, &c. and consequently that a is at.

240. We have also shown, that to find the cube of any power, as a", we have only to multiply its exponent by 3, and that we shall have as" for the cube. Conversely, therefore, in order to find the cube root of a3", we have only to divide the exponent by 3, and we shall obtain a" for the root required; and consequently that a will be the cube root of a3; ao of a; a3 of ao, and so on.

We

We have therefore no difficulty in applying these principles to cases where the index is not divisible by 3, or in concluding that the cube root of a2 is a3; of a1, a31; or generally

n

that the cube root of a" is a. Consequently that a may always be represented as a+

241. The same principles will likewise apply to powers of a higher degree; thus the fourth root of a, ora, may be expressed a; 5/a, a3, and generally the nth root of a, a", or

1

242. These methods of representing the powers and roots of quantities will be found to be of the most essential use in algebraical calculations, and require very particular attention, since by means of them we are enabled to reject altogether the radical signs that have been hitherto made use of, and by substituting in their stead the indices or exponents of the powers, whether integral or fractional, to proceed to the calculation of surds and irrational quantities with the same facility as of any other species of numbers..

243. We are further enabled by the same means to reduce reciprocally rational quantities to the form of surds, for since we have seen that a* is equivalent to /a, it is evident that any rational quantity as a may be reduced to the same form by first raising it to that power which is denoted by the root of the surd, and then annexing to it the radical sign. Thus if it were required to reduce 8 to the form of a, we have only to write 64, which we know to be equivalent to 8.