(8) 4 abx-3 cdy+2z, 5 abx+2cdy-4z, -7 abx+4 cdy -6z, 9 abx-5 cdy—z.

(9) 15 mxy+7 nz-3 ab, -11mxy-13 nz+ab,—mxy+ nz-2 ab, -25 ab.

(10) 10xyz-3abc+5de, 3xyz-5abc-3de, —7xyz— 2abc, xyz+de.

(11) -2cx+3 dy+4 xz-8, 6 cx-7 dy-5x2+14, -9 cx +dy-6, 4 dy-8.

(12) 5x, 7x+4 aby, 3 x-9 aby+2 cz, -x+aby-cz+ 7, 3 cz- -12.

(13) -4 yz, 2 ax+6 yz-2 bc, 3 yz-7 bc, 5 ax-2yz+ 6 bc, -13 ax+7yz+19 bc, ax-yz-bc, 15 ax-11yz+ 3 bc, 5 ax+yz-4 bc.

(14) 3abx-7dy+ez,-8 dy-3 ez, 4 abx+6ez, 8 dyez,-5 abx+2dy-3ez, 2abx +4 dy + 2ez, 4 ez-5 abx.

16. When the quantities to be added are not all like quantities.

RULE. Add together those of the quantities that are like, by the rule for the addition of like quantities; and to the sum connect, with their proper signs, the remaining unlike quantities.

NOTE. In the following examples, illustrative of this rule, the quantities, although placed in vertical columns, will not be so arranged as that the like terms may stand all under one another, as in the vertical rows summed in the preceding article. The learner, therefore, will have to examine all the columns, even at the outset, in his search for those quantities, like the one first chosen to begin with. It is of no consequence which set of like quantities be summed first, but it is the custom of algebraists to commence with the quantity at the top of the first column, and to put down, under that column, the sum of all the quantities like it; then, to go to the quantity at the top of the second column, and so on. And this custom we shall in general follow.

12bz-12cd+5 mn ef +7

(2)
4xy-3 az+6
-6az+4-2xy
14-xy+az
5xy -az-7
2az+13-3d

6xy-7 az+30—3 d

(4)
6ab3c+5 de

-7 de +4 ab +3c
8c-2 de ab
2x+3ab12
-9ab-8c +f
-3f +ab -c
6ab-17c-4 de-2ƒ—12

8az+2cd-2k + 13ef+3p-2q

In the first of these examples, there are 2x and 7x in the first column; Dr in the second column; and -3x, 5x and —4x in the third column:

the sum of the like quantities is 5x. Again, in the second column there are 3y and -6y; in the first 4y and 4y; and in the third no like of these quantities: the sum of the second set of like quantities is therefore 5y. Lastly, in the third column there area and 4a; in the first, 2a; and in the second, 5a: the sum of these is 10a. The whole sum of the like quantities is, therefore, 5x+5y+10a, to which the quantity 36, which has no like, must be connected with its proper sign plus. The other sets of expressions are summed in a similar manner.

(6). The sum of 2x-3y+5az, 4y-2az-x, 7az+5x-4y, 3x+bc, 5y-bc-3x, is 6x+2y+10az.

(7). The sum of 9 ab+8c+2de, 8c-ab-f, 3f-ab+c, -2x-3 ab, 7 de-4 ab, 3c, 3c-5de, is 17c+4 de+2f. (8). The sum of ax+2by-cz, 3 cz-4d, 24ax-4cz, 5d+e, 6cz-a, 7by—ax+d, is 2ax+9by+4cz+2d—a+e. It is evident that sets of expressions like these may be added together quite as conveniently as they thus stand, as if they were arranged under one another, so as to present rows of vertical columns.

(7) 4 axy-2bxz+3c, 5 ab +7c, -6bxz-13axy+ab, 2axy-c, 7bxz+2axy-5c, 6c-4 d.

(8) 2px-3y+rs, -az-5 px-4qy, 6rs+5 az-11, -4 px+3,-7 qy+px-8rs, a+17, 3p-4q+2 az.

(9) 4axy-7bz+3c, 5x-7y+8c-4ab,-c+3x-5axy+2bz, 6bz+4y-7x+3axy-7c, 9x-13y+1.

These examples of addition of algebra will help to familiarise the learner with the use of the plus and minus signs, and with the method of reducing lengthy compound expressions to a more simple form whenever, by the occurrence of like quantities, such reduction is possible. We shall now resume the definitions, and explain some additional particulars respecting algebraic notation.

DEFINITIONS CONTINUED.

17. When quantities or factors are multiplied together, the result is called a product, but if the factors are all equal, the product is then called a power. There is a very convenient notation for powers: instead of writing the factors one after the other, as we do in the multiplication of unequal factors, merely one of the equal factors is written, and the figure expressing the number of them is placed, in smaller type, a little above it to the right. Thus the product of the five factors a, a, a, a, a, would not be written aaaaa, but simply so, a5. In like manner, the product of the three factors x, x, x, would be written ; that is, the factor to be repeated is written down only once, and the number of repetitions of it is expressed by the small figure at the upper corner. The different powers take their names from the value of this corner figure; thus x2 is called the second power of x, or more frequently, the square of x; x3 is called the third power, or the cube of x; x is the fourth power of x; x5 the fifth power of x; as the eighth power of a; and so on. If a stand for the number 2, that is

If x=2, then a2 —4, x3—8, x1—16, x5—32, x6—64, x1=128, x8=256, &c.

If x=3, then a2-9, x3-27, x1-81, x5-243, x6-729, &c. If x=1, then x2=1, x3—1, ∞a—1, x3—1, x¤—1, &c.

and so on.

18. These small corner figures are called exponents or indices: the exponent or index for the square of a quantity is 2; the exponent or index for the cube is 3; that for the sixth power is

6; and so on. The learner will therefore now be able to interpret such forms as 23, 32, 43, &c. the first stands for 8, the second for 9, the third for 64, &c. He must be very careful not to confound coefficients with exponents, and to avoid calling aa, four times a, b3, three times b, &c. four times a is represented by 4a, three times b by 3b, &c. 4 a means that four a's are to be added, a1 means that they are to be multiplied.

19. The quantity, which by repeated multiplication by itself produces a power, is called a root of that power; thus, 2 is the cube root of 8, 3 is the square root of 9, 4 is the cube root of 64, &c. And generally any quantity whose square, cube, fourth power, &c. produces another quantity, is called the square root, cube root, fourth root, &c. of that other quantity. To find any proposed power of a given quantity is, in general, an easy operation, as it requires only repeated multiplications by the same thing. But to find a proposed root of a given quantity, that is, to perform the reverse operation, is often a matter of considerable difficulty, as the learner will be prepared to expect from his recollection of the processes for the square and cube roots of numbers in arithmetic.

The notation, however, for the roots of quantities is nearly as simple as that for their powers. It consists in prefixing to the quantity proposed the symbol, which is only a corruption. of the first letter in the word root, and placing a little above it an index to mark what root is meant; thus, a means the square root of a, a means the cube root of a, a the fifth root of a, and so on; but as there is no root lower than the second or square root, when that is to be represented it is considered sufficient to write the symbol without any index, so that, in strictness, the square root of a would be written simply a, for here, as in all other cases, algebraists take care to suppress useless marks and symbols, and simplify their notation as much as possible.

But there is another, and, in general, a still more convenient notation for roots, a notation more strictly in keeping with that for powers, the proper exponents or indices being placed over the right hand corner of the quantity proposed, and the symbol ✔dispensed with; thus,

mean the same as

a1, a3, a3, x1, y3, &c.

Na, a, a, x, y, &c.