SUBTRACTION. RULE. Change the signs of the quantities to be Subtracted, or conceive them to be changed; and collect them together as in Addition. * In example second, 2a, which is to be subtracted from 3a, is conceived or called-2a, then by case second in Addition, their difference will be a. Also, +3x, which is to be subtracted from -2r, is conceived or called -3x, then by the same case their difference is -5x. 4. From 25-3x take one quarter of that quantity. 75-9x Ans. 4 MULTIPLICATION. MULTIPLICATION of Algebra, as in common Arithmetic, is finding the product of two or more quantities, and is commonly divided into three cases. CASE I. When the quantities are both simple. RULE.-Multiply the coefficients of the quantities together, and annex all the letters in both quantities, or their powers: prefixing the sign + or to the product, according as the quantities have like, or unlike signs. In Multiplication, like signs produce +, and unlike signs -, that is, if the multiplicand and multiplier be both +, or both the product will be +; but if one be +, and the other the product will be minus, see notes 3rd and 4th. 1. In the 1st example both the quantities are +, therefore, the product is plus, and is performed as follows, viz. 2×3=6, and axxax, hence ar being annexed to the 6 makes the product 6ar. 2. In the 3rd example, 3×4=12, and axaa therefore the product is 12a2; a' being the same as a, every letter, without an index, is supposed to have 1 for an index, hence, multiplying the same letters or their powers together is nothing more than adding their indices together. Thus, a×a=a1×a1=a1+1—a2; x2×x=x2+1=x3; x3×x2 3. When minus and plus quantities are to be multiplied together, as in the 4th example, viz. —2×+4 makes -8, it being the same thing, as adding four times-2 together, which will make -8. 4. In the 5th example both the quantities are minus; then since 3x-3x=0, .. (3x—3x) ×—2â——6ax+6ax=0, because O, multiplied by any quantity, produces nothing; and since the 1st term of the product, viz. 3xx-2a, (by note 3) makes -6ax; therefore the 2nd product, viz. -3x X-2a must make 6ax, otherwise the whole product -6ax+6ax cannot be equal to nothing. B When one of the quantities is a compound, and the other a simple quantity. RULE.-Multiply each term of the compound quantity by the simple quantity, beginning at the left hand or first term of the compound quantity, and proceeding to the right hand or last term; placing the quantities or products one after another with their proper signs. When both the factors are compound quantities. RULE.--Multiply each term of the multiplier, into every term of the multiplicand, beginning at the left hand and proceeding to the right, as in case 2nd. setting down these products one after another with their proper signs; then add them together by the different cases in Addition. Surd quantities, under the same radical sign, are multiplied as rational quantities; only the product must be placed under the same radical sign. Thus 9×√7=√9x7=√/63, √ax √x=√√ax,√/4a2x2×√/6ax=√/24a3r3, &c. see example fourth which is solved. |