Find (by either method) the cash balance in the following bills, reckoning interest at 6%: NOTE. Since the Dr. items have the same term of credit, find the equated time of these items, and count forward from that date 3 mos. for the term of credit. Jan. 3. To mdse. 30 dys. $100.00 || Feb. 25. By cash, CR. $100.00 CHAPTER XXI. POWERS AND ROOTS. 378. The square of a number is the product of two factors, each equal to this number. are Thus, the squares of 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100. 379. The square root of a number is one of the two equal factors of the number. are Thus, the square roots of 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. 380. The square root of a number is indicated by the radical sign, or by the fractional exponent above and to the right of the number. written 381. Since (24)2=(16)2=256=28; and (2a)* = (16)* = =22, it is evident that, A power of a power of a number, or a root of a power of a number, is that number with an exponent equal to the prod uct of the given exponents. 382. Since 35=30+5, the square of 35 may be obtained as follows: 383. Hence, since every number consisting of two or more figures may be regarded as composed of tens and units, The square of a number will contain the square of the tens twice the tens X by the units + the square of the units. SQUARE ROOT. 384. The first step in extracting the square root of a number is to mark off the figures in periods. Since 112, 100 = 102, 10,000 = 1002, and so on, it is evident that the square root of any number between 1 and 100 lies between 1 and 10; of any number between 100 and 10,000 lies between 10 and 100. In other words, the square root of any number expressed by one or two figures is a number of one figure; of any number expressed by three or four figures is a number of two figures; and so on. If, therefore, an integral number be divided into periods of two figures each, from the right to the left, the number of figures in the root will be equal to the number of periods. The last period at the left may consist of only one figure. Ex. Find the square root of 1225. 12'25 (35 9 65)3 25 3 25 Since 1225 consists of two periods, the square root will consist of two figures. The first period, 12, contains the square of the tens' number of the root. The greatest square in 12 is 9, and the square root of 9 is 3. Hence, 3 is the tens' figure of the root. The square of the tens is subtracted, and the remainder, 325, is twice the tens X the units + the square of the units. Twice the 3 tens is 6 tens, and 6 tens is contained in the 32 tens of the remainder 5 times. Hence, 5 is the units' figure of the root. Since twice the tens X the units + the square of the units is equal to (twice the tens + the units) × the units, the 5 units are annexed to the 6 tens, and the result, 65, is multiplied by 5. 385. The same method will apply to numbers of more than two periods, by considering the part of the root already found as so many tens with respect to the next figure of the root. Ex. Extract the square root of 7890481. 7'89'04'81 (2809 4 48) 3 89 When the third period, 04, is brought down, and the divisor, 56, formed, the next figure of the root is 0, because 56 is not contained in 50. The O is then placed both in the root and the divisor, and the next two figures, 81, are brought down. 386. If the square root of a number have decimal places, the number itself will have twice as many. Thus, if .11 be the square root of some number, the number will be (.11)2= .11 x .11 = .0121. Therefore, the number of decimals in every square number will be even; and the number of decimal places in the root will be half as many as in the given number itself. Hence, if a given square number contain a decimal, and if it be divided into periods of two figures each, by beginning at the decimal point and marking toward the left for the integral number, and toward the right for the decimal, the number of periods to the left of the decimal point will show the number of integral places in the root, and the number of periods to the right will show the number of decimal places in the root. Ex. Extract the square root of 52.2729. 52.27'29 (7.23 49 142)3 27 284 1443) 43 29 It will be seen from the periods that the root will have one integral and two decimal places. 43 29 387. If a number contain an odd number of decimal places, or if any number give a remainder, when as many figures in the root have been obtained as the given number has periods, then its exact square root cannot be found. We may, however, approximate to the exact root as near as we please, by annexing ciphers and continuing the operation. The result will be a constantly varying succession of figures. Ex. Extract to six places of decimals the square root of 19. 388. The square root of a common fraction is found by extracting the square roots of the numerator and denominator. But when the denominator is not a perfect square, it is best to reduce the fraction to a decimal and then extract the root. |