.. to the denominator of the second; and the denominator of the second, minus twice the numerator of the first, is equal to the numerator of the second. What are the fractions? Ans. and . 12. A man bought a horse, a wagon, and a harness, for $180. The horse and harness cost three times as much as the wagon, and the wagon and harness one half as much as the horse. What was the cost of each? 13. A gentleman gives $600 to be divided among three classes in such a way that each one of the best class is to receive $10, and the remainder to be divided equally among those of the other two classes. If the first class proves to be the best, each one of the other two classes will receive $5; if the second class proves to be the best, each one of the other two classes will receive $4; but if the third class proves to be the best, each one of the other two classes will receive $2. What is the number in each class? 14. A cistern has 3 pipes opening into it. If the first should be closed, the cistern would be filled in 20 minutes; if the second, in 25 minutes; and if the third, in 30 minutes. How long would it take each pipe alone to fill the cistern, and how long would it take the three together? Ans. 1st, 85 minutes; 2d, 46 minutes; 3d, 35 minutes. The three together, 16 minutes. 15. Three men, A, B, and C, had together $24. Now if A gives to B and C as much as they already have, and then B gives to A and C as much as they have after the first distribution, and again C gives to A and B as much as they have after the second distribution, they will all have the same sum. How much did each have at first? Ans. A, $13; B, $7, and C, $4. SECTION XVI. POWERS AND ROOTS. 119. A POWER of any quantity is the product obtained by taking that quantity any number of times as a factor; and the exponent shows how many times the quantity is taken (Art. 24). Thus, aa1 is the first power of a; 120. In order to explain the use of negative indices, we form, by the rules of division, the following series: We form the first series as follows: a divided by a gives a*; a1 by a, gives a3; a3 by a, gives a2; a3 by a, gives a; a by a, gives a 1 1 a2; a2 by a, gives and so on. 1; 1 by a, gives; by a, gives The second series is formed in the same way from a to a; but if we follow the same rule of division from a toward the right as from a3 to a, viz. subtracting the index of the divisor from that of the dividend, a divided by a, gives ao; a° by a, gives a-1; read a, with the negative index one; a-1 by a, gives a-2; a-2 by a, gives a→3; and so on. From this we learn, 1st. That the 0 power of every quantity is 1; 2d. That a-1, a-2, a-3, &c., are only different ways of Any two quantities at equal distances on opposite sides of ao, or 1, are reciprocals of each other. 121. The rules given for the multiplication and division of powers of the same quantity (Arts. 50 and 54) apply equally well whether the exponents are positive or negative. For The following examples în multiplication are to be done according to the rules for the multiplication of powers of the same quantity by each other, given in Art. 50; and those in division, by the rule for the division of powers of the same quantity by each other, given in Art. 54. 6. Multiply 4x-3y-5 z by 3 x3 y5 za. -3 16. Divide 4 x2y-z by 2 ax-2y-3 z2. 17. Divide 7 a2 bx4ys by 10 a b-1x4y-3 22. -7 Ans. y2. Ans. a-5 b5 c3. 18. Divide 144 a2 b c 2 x2 y z by 16 a2b-c2x2y. 122. It follows from the preceding article that a factor may be transferred from the numerator of a fraction to its denominator, or vice versa, provided the sign of the exponent of the factor is changed from + to, or to +. For 123. INVOLUTION is the process of raising a quantity to a power. 124. A quantity is involved by taking it as a factor as many times as there are units in the index of the required power. 125. According to Art. 48, (+a) x (+a) × (+a)=(+a2) × (+a)=+a3, and so on; and (+a) x (+a)=+a2, (− a ) × (—a)=+a2, (-a) X (-a) × (-a) = (+a2) × (-a)=-a3, (-a) X (-a) X (− a) × (− a) = (—a3) × (− a)=+a*, and so on. Hence, for the signs we have the following RULE. Of a positive quantity all the powers are positive. Of a negative quantity the even powers are positive, and the odd powers negative. |