These five Examples require but little explanation, as all relative to them will appear evident, by strict attention to the adding of the indices of similar quantities. In the 14th Example, the index of a ist, and the index of a is 1, therefore, by adding these indices, we have 11 = , for the index of the product of a multiplied by a; and then, multiplying ✔ by a, we have avb. Next, multiplying by the quantity b, the product will be ✔ab + b3, and the signs of the multiplier and multiplicand being similar, each product requires an affirmative sign; therefore, by Addition, the whole result will be a2 + ab+b2, which is the same with a3 + Examples 12 and 13 are similarly worked. ab2 + 3 a√b + √ ab + √ 63. By adding the indices in the 15th example, it is evident we have 2n for the index of a. In this, and the 16th example, we have the solution of the Geometrical Theorem, given as the 3rd Corollary of the 5th Proposition of the Second Book. It is evident that n 1 added to n 1 becomes 2n - 2, for the index of a; and the indices of the other products may be similarly found. In these two Examples we have the proof of the 4th Proposition of the Second Book, a2 and b2 being the squares of the parts, and 2ab equal to twice the rectangle under the parts. cb) a3 + (b4 c2)a3 + (b1 — c1)a2 + 3(c + b)a bc3) a2 + 3ab (bc — c2) a3 + (b3c — b2c2 + bc3 — c1)a2 + 3ac b3c + b2c2 Here, as in Addition and Subtraction, we find the advantage of a parenthesis, to enclose all those quantities which are multiplied by the same co-efficient; it is evident that the work would otherwise be much longer, as the above Answers written at length would be the following: 19th Answer....x6 + bx3 + cx3 acx2 + bc. a2x4 abx2+ Ans.....a — 2y + a¬yb−x + a −yb −2x + b 3.x 2.-Multiply a3b + a1 + a3b2 — a1b + a1b2 by a3 — a2b2 Ans...(1 (1 + 63)a7. a3b + a2b. b2)b2a5 (4 — b + b2)b2a6 3. Multiply 3x2y2-2xy + 6 by bxy. Ans...3bx3y3 -2bx2y2+6bxy. Ans...3bqam+n — 4bram + y + 5qran+y+ b2a2m+2q2a2n + 3r2a2y It will be found essential to bear in mind the nature of the indices, and then the student can advance with ease to himself. By referring to the 12th Definition, (given at the end of 1st Book) you will see the properties of the 1st Example in this Case; but it may be further added, that a y indicates the nought power always indicates unity; therefore a - y = = y leaves the nought power of a, it is evident that a ao ay 1 ау Similarly 2a-y means 2 ay and 6 x 5 2 means 6 = We shall again comment on this topic, 52 25 under the head of Roots. DIVISION. RULE I. Arrange the quantities in proper order, as in Arithmetic; and, this Rule being the reverse of Multiplication, we subtract the indices of similar letters, and place the others in the quotient, as we find them in the dividend. = a3, because by subtracting the index Ex. gr.: a4 a of the divisor from that of the dividend, there will remain for the quotient, a3; and a2d3 a = ad3; it is also to be understood that numerical co-efficients are to be divided, as in Arithmetic, and that like signs produce an affirmative result, but unlike signs produce minus. EXAMPLES WITH ANSWERS. 1.-Divide 12a6x8 by 4a2x2. |