OPERATION. 511. When the quantity and rate of a mixture, with the rates of the articles composing it, are given, to find the quantity of each article which is not limited. Ex. 1. How many gallons of water must be mixed with wine at $1.50 a gallon so as to make a mixture of 100 gallons worth $ 1.20 a gallon ? Ans. 20 gallons. Representing the 100 Slgal. at 0.00, gain 1.20 1.20 rate of the water by . Igal. at 1.50, loss .30) 0.00, we then find, ļ1.20 as in Art. 509, the 3gal. at 1.50, loss .90) quantity required of each article, in com1+4= 5gal. ; } of 100gal. = 20gal. Ans. posing a mixture of the given mean, to be 1 gallon of water and 4 gallons of the wine. Therefore, the quantity of water is to the whole quantity of. the mixture as 1 to 5. Hence, in a mixture of 100 gallons at the mean rate given, the water must be of 100 gallons, or 20 gallons. RULE. — Find the proportional quantities of the several articles, as in Art. 509, or 510, as though the quantity of the mixture were not limited. Then take such a part of the given quantity of the mixture, as each of these proportional quantities is of their sum. MPLES. 2. A merchant has sugar at 8 cents, 10 cents, 12 cents, and 20 cents a pound; with these he would fill a hogshead that would contain 200 pounds. How much of each kind must he take, so that the mixture may be worth 15 cents a pound? Ans. 33}lb. of 8, 10, and 12cts., and 100lb. of 20cts. 3. How much wheat at $ 2.00 and $ 1.80 a bushel must be mixed with 4 bushels at $ 2.20, and 10 bushels at $ 1.70, so as to make a mixture of 48 bushels, worth $ 1.90 per bushel ? Ans. 21 bushels $ 2.00; 13 bushels at $ 1.80. 4. How much gold of 15, 17, and 22 carats fine must be mixed with five ounces of 18 carats fine, to make a composition of 5 pounds, that shall be 20 carats fine ? 5. A gentleman's servant having been ordered to purchase 20 animals for $ 20, brought home sheep at $ 4.00, lambs at $0.50, and kids at $ 0.25 each. Required the number of each kind. Ans. 3 sheep; 15 lambs; and 2 kids. MISCELLANEOUS EXAMPLES. of 3• Exchange be per cent. premiu of Live 1. A manufacturer employs a number of men at $ 1.20, and a number of boys at $ 0.80, per day; and the amount of the wages of the whole is the same as if each had $ 0.974 per day. Required the number of men, that of the boys being 9. Ans. 7 men. 2. What is the value of 5000 specie rix dollars 12 skillings of Sweden in United States money? Ans. $5300.265. 3. Exchange between New Orleans and England being in New Orleans at 8 per cent. premium, and in Liverpool at 10 per cent. premium, if L. Sandford of Liverpool owes M. Lassale of New Orleans for cotton to the amount of 1500£. 15s. sterling, what will be the difference between Lassale drawing or Sandford remitting the amount ?. 4. If 17 gallons of spirits at $ 1.26 per gallon be mixed with 7 gallons at a different price, and 20 per cent. be made by selling the mixture at $ 1.56, what was the price of the latter kind per gallon ? Ans. $ 1.39 per gallon. 5. What is the value of 100 ounces 20 tari 10 grani of Sicily in lire and centesimi of Leghorn ? Ans. 1510 lire 25 centesimi. 6. If 20 United States gallons equal 1 eimer of Sweden, 3 eimers of Sweden equal 4 eimers of Trieste, 24 eimers of Trieste equal 9 ahms Danish, and 33 ahms Danish equal 5 carri of Naples, which will cost the most in United States money, 170 eimers of Trieste of wine at 1 florin 45 kreutzers per gallon, or 12 carri of wine at 1 ducat of Naples a gallon ? 7.0 When exchange on England is at 8 per cent. premium, and freight at 12d. per United States bushel, how much can be paid per bushel for wheat in Baltimore, in answering an order from Liverpool limited to 60s. per imperial quarter? Ans. $ 1.500 per bushel. 8. A merchant mixes 11 pounds of tea with 5 pounds of an inferior quality, and gains 16 per cent. by selling the mixture at 87 cents per pound. Allowing that a pound of the one cost 12 cents more than a pound of the other, what was the cost of each kind per pound ? Ans. The one 78 cts. ; the other 66cts. per lb. INVOLUTION 512. INVOLUTION is the process of finding the powers of quantities. A power of a number or quantity is the result obtained by taking that quantity a certain number of times as a factor. 513. The number from which a power is derived is called the root of that power. The first power is the root, or the number involved. The second power is the product of the root multiplied by itself once, or used twice as a factor. The third power is the root used three times as a factor; &c. 514. The index or exponent of a power is a small figure written at the right, above the root, indicating the number of times it is employed as a factor. Thus, the second power of 4 is written 42, the third power of 9 is written 93, and the fourth power of his written (*). NOTE. — In denoting the power of a fraction, the fraction is included in a parenthesis, in order that the exponent may be regarded as applying to the whole expression, and not to the numerator alone. When no index is written, the number itself is to be considered the first power. The second power is sometimes called the square of a number, the third power the cube, and the fourth power the biquadrate. 515. To raise a number to any required power. 2 = 2, the first power of 2, is written 21 or 2. 2X2 = 4, the second power of 2, is written 22. 2X2X2 = 8, the third power of 2, “ “ 23. 2x2x2x2 = 16, the fourth power of 2, “ " 24. 2x2x2x2x2 = 32, the fifth power of 2, “ " By examining the several powers of 2 in the examples, it is seen that each has been produced by taking the 2 as a factor as many times as there are units in the exponent of each power raised. Hence the Rule. — Multiply the given number into itself, till it has been used as a factor as many times as there are units in the exponent of the power to which the number is to be raised. NOTE 1. — The number of multiplications will always be one less than the number of units in the exponent of the power to be raised, since in the first 23. multiplication the root is used twice; once by being taken as the multiplicand, and once more as the multiplier. Note 2. – A fraction is involved by involving both its numerator and its denominator. EXAMPLES. 1. What is the 3d power of 8 ? Ans. 512. 2. What is the 5th power of 4 ? Ans. 1024. 3. What is the 3d power of ? Ans. . 4. What is the 4th power of 23? Ans. 5041 5. What is the 5th power of } ? Ans. 2 3 6. What is the 6th power of 5 ? Ans. 15625. 7. What is the 6th power of 13 ? Ans. 1613114 8. What is the value of 710 ? Ans. 282475249. 9. What is the value of .0454 ? Ans. .000004100625. 516. To raise a number to any required higher power, without producing all the intermediate powers. Ex. 1. What is the 7th power of 5 ? Ans. 78125. OPERATION 3 + 2 + 2 5, 25, 1 35; 125 $ 25 25 378 1 25. We raise the 5 to the 2d and to the 3d power, and write above each power its exponent. Then, by adding the exponent 2 to itself, and increasing the sum by the exponent 3, we obtain 7, a number equal to the exponent of the required power; and by multiplying 25, the power belonging to the exponent 2, into itself, and the product thence arising by 125, the power belonging to the exponent 3, we obtain 78125, the required 7th power. Therefore, The product of two or more powers of the same number is that power which is denoted by the sum of their exponents. Hence, the RULE. — Multiply together two or more powers of the given number, the sum of whose exponents is equal to the exponent of the power required, and the product will be that power. NOTE. — When the number to be involved contains a decimal, it is generally sufficient to retain in the result not more than six places of decimals; and the werk may be accordingly contracted as in the multiplication of decimals (Art. 273). EXAMPLES. 2. What is the 7th power of 8 ? Ans. 2097152. Ans. 40353607. 5. What is the 5th power of 195 ? Ans. 281950621875. 6. What is the 6th power of ? Ans. 7. Required the 2d power of 4698. 8. Required the 2d power of 6031. Ans. 36372961. 9. What is the 13th power of 7 ? Ans. 96889010407. 10. What is the 12th power of 6 ? 11. What is the 15th power of 9? Ans. 205891132094649. 12. What is the 4th power of 4.367 ? Ans. 363.691179+. 13. Involve the following numbers to the powers denoted by their respective exponents : (23)5, 1.04', and (34)*. Ans. 157 28230; 1.800943+; 1161331. EVOLUTION. 517. EVOLUTION, or the extraction of roots, is the process of finding the roots of quantities. It is the reverse of involution. 518. The root of a quantity or number is such a factor as, being multiplied into itself a certain number of times, will produce that quantity or number. The root takes the name of the power of which it is the correlative term. Thus, if the number is a second power, the root is called the second or square root; if it is the third power, the root is called the third or cube root; if it is the fourth power, its root is called the fourth or biquadrate root; and so on. Rational roots are such as can be exactly obtained. Surd roots are such as cannot be exactly obtained. 519. Roots are usually denoted by writing the radical sign, ✓, before the power, with the index of the root over it; in case, however, of the second or square root, the index 2 is omitted. Thus, the third root of 27 is denoted by 27, the second root of 16 is denoted by V16, and the fourth root of 1 is denoted by 1. |