EXAMPLES. 1. In 247cwt. 2qrs. 15lb. grofs; tare 28 per cwt. and tret 4lb. per 104lb.; what neat weight? lb. Cwt. qrs. lb. 28247 2 15 grofs. 61 3 17 12 tare, fubtract. 2. What is the neat weight of 4hhds. of tobacco, weighing as follows; the 1ft, 5cwt. 1qr 12lb. grofs; tare 65lb. per hogshead; 2d, 3cwt. cqr. 19lb. grofs ; tare 75lb.; the 3d. 6cwt. 3qrs. grofs; tare 49lb.; and the 4th, 4cwt. 2qrs glb. grofs; tare 35lb. and allowing tret to each as ufual? Anf. 17cwt. ogr. 19lb. CASE IV. When tare, tret and cloff are allowed ;-Deduct the tare and tret as before, and divide the futtle by 168, and the quotient will be the cloff, which fubtract from the futtle, and the remainder will be the neat. EXAMPLES. 1. What is the neat weight of a hogfhead of tobacco, weighing 16cwt. 2qrs. 20lb. grofs; tare 14lb. per cwt-tret 4lb. per 104, and cloff 2lb. per 3cwt. ? 4lb. is)14 2 10 8 O 2 6 13 tret, fubtract. 2lb. is 8) 14 is)14 o 3 11 futtle. 95 cloff, fubtract. Anf. 13 3 22 6 neat. 2. If ghhds. of tobacco, contain 85cwt. oqr. 2lb. tare, 30lb. per hhd. tret and cloff as ufual; what will the neat weight come to, at 6d per lb. after deducting, for duties and other charges, 511. IS. 8d. ? Anf. 1871. 18s. 5d. INVOLUTION, OR, TO RAISE POWERS. A Power is the product arifing from multiplying any given number into itfelf continually a certain number of times, thus: 3X39 is the 2d power, or fquare of 3.32 3X 3X3=27 is the 3d power or cube of (3.=33 3X3X3X3=81 is the 4th power, or the biquad(rate of 3, &c. 34 The number denoting the power is called the Index, or the Exponent of that power; thus, the fecond power of 5 is 25, or 52, &c. 2 X 2-4, the fquare of 2: 4X4=16=4th power of 2: 16× 16=256=8th power of 2, &c. RULE. Multiply the given number, root, or first power continually by itself, till the number of multiplications be one lefs than the index of the power to be found, and the laft product will be the power required. Note. Whence, becaufe fractions are multiplied by taking the products of the numerators, and of their denominators, they will be involved by raising cach of their terms to the power required; and if a mixed number be propofed. either reduce it to an impr oper fraction, or reduce the vulgar fraction to a decimal, and proceed by the rule. A Surfolids, Squared Third Surfolids, 84475249 07741824 486784401 1011024 59049 1048576 9765625 60466176 Square Cube, Squared ; 12 14096 53144116777216 244140625 217678233613841287201 687 1 9 4 7 6 7 36 28 2429536481 The Root is a number whofe continual multiplication into itself produces the power, and is denominated the fquare, cube, biquadrate, or zd, 3d, 4th root, &c. accordingly, as it is, when raifed to the 2d, 3d, 4th, &c. power, equal to that power. Thus, 4 is the fquare root of 16; becaufe 4X416, and 3 is the cube root of 27, because 3×3×3=27; and fo on. Although there is no number of which we cannot find any power exactly, yet there are many numbers, of which precife roots can never be determined. But by the help of decimals, we can approximate towards the root, to any affigned degree of exactness. The roots, which approximate, are called furd roots; and thofe which are perfectly accurate, are called rational roots. Roots are fometimes denoted by writing the charafter before the power, with the index of the root over it; thus the 3d root of 36 is expreffed 36, the index 2 being omitted when the fquare root is defigned. 3 36, and the 2d root of 36 is If the power be expreffed by feveral numbers, with the fignor between them, a line is drawn from the top of the fign over all the parts of it; thus, 3 the 3d root of 47 + 22 is No 47 +22, and the 2d root of 59 17 is 59-17, &c. Sometimes roots are defigned like powers, with fractional indices; thus, the fquare root of 15 is 151, the cube root of 21 is 213, and the fourth root of 3720 is 37--204, &c. The EXTRACTION of the SQUARE ROOT. RULE 1.-Diftinguish the given number into periods of two figures each, by putting a point over the place of units, another over the place of hundreds, and fo on, which points fhew the number of figures the root will confift of. 2. Find the greateft fquare number in the first, or left hand period, place the root of it at the right hand of the given number, (after the manner of a quotient in 'divifion) for the first figure of the root, and the fquare number, under the period, and fubtract it therefrom, and to the remainder bring down the next period for a diyidend. 3. Place the double of the root, already found, on the left hand of the dividend for a divifor. 4. Seek |