Table of the SQUARES and CURES of the nine digits. Roots. 1 2 3 41 5 6 7 8 9 NOTE.-The num ber denoting the height of the power, is called the index, or exponent of that power: so the 2d power of 3 may be denoted by 32, the 3d by 33, the 4th by 34, &c.; the 6th power 4096 4th power, or biquadrate. of 8 by 8°, &c. 8 32768=5th power, or sursolid. 8 Ans. 8*. Ans. 128 Ans. 59049. 2187 262144-6th power, or square cube. 2. What is the 7th power of ? 3. What is the 5th power of 4. What is the fourth power of,27? Ans. ,00531441. or 11? 7776. EVOLUTION, OR EXTRACTION OF ROOTS. WHEN the root of any power is required, the business of finding it is called the extraction of the Root. The root is that number, which by a continual multiplication into itself, produces the power which is given to be extracted. Though every number will produce a perfect power by involution, yet there are many numbers, the precise roots of which can never be determined. By the help of decimals, however, we can approximate towards the root, to any assigned degree of exactness. The roots which approximate, are called surd roots, and those which are perfectly accurate, are called rational roots. TO EXTRACT THE SQUARE ROOT. Any number multiplied into itself, produces a square. The extracting of the square root, is only finding a number, which, being multiplied into itself, shall produce the given number. RULE.-1. Distinguish 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 so on over every second figure; and if there be decimals, point them in the same manner, from units towards the right hand; which points show the number of figures the root will consist of. 2. Find by the table or trial the greatest square 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 division,) for the first figure of the root; and set the square number under the period, subtract it therefrom, and to the remainder bring down the next period for dividend. 3. Place the double of the root already found on the left hand of the dividend for a divisor. 4. Consider what figure must be annexed to the divisor, so that if the result be multiplied by it the product may be equal to, or the next less than, the dividend, and it will be the second figure of the root. 5. Subtract that product from the dividend, and to the remainder bring down the next period for a new dividend. 6. Find a divisor as before, by doubling the figures already in the root; and from these find the next figure of the root as in the last article; and so on through all the periods to the last. Or, to facilitate the foregoing operation, when a period is brought down to a remainder, and a dividend thus formed, in order to find a new figure in the root, divide said dividend, (omitting the right hand figure thereof,) by double the root already found, and the quotient will commonly be the figure of the root sought, or, being made less by one, or two, will generally give the next figure sought, TO EXTRACT THE SQUARE ROOT OF A VULGAR FRACTION. First prepare all vulgar fractions by reducing them to their lowest terms, both for this and all other roots. Then, 1. Take the root of the numerator and that of the denominator for the respective terms of the root required; and this is the best way if the denominator be a complete power. But if not, 2. Multiply the numerator and denominator together; take the root of the product; this root, being made the numerator to the denominator of the given fraction, or the denominator to the numerator of it, will form the fractional root required. 3. Or reduce the vulgar fraction to a decimal, and extract its root. EXAMPLES. 1. Required the square root of 6749604. 6749604(2598 Ans. The root exactly without a 4 45)274 509)4996 9)4581 remainder; but when the periods belonging to any given number are all exhausted, and still leave a remainder, the operation may be continued at pleasure, by annexing periods of ciphers, &c.-Roots are often denoted by writing before the power, with the index of the root within or over it, save the index of the square root, which is ever understood: so✓ 64 is the 2d or square root of 64; 2. Required the 64 the 3d or cube root of 64; *✔✅ 64 square root of 739,4. the 4th root of 64. 5188)41504 8)41504 739,40(27,19+root. 4 47)339 541) 1040 541) 541 4 When the square root of a number is wanted to many places, the work may be much abridged. Find half the root by the rule; then to get the rest, annex to the last remainder as many ciphers as you need, and divide it by the double of the root before found. 3. What is the square root of 2? Ans. 1,41421356.+ 4. What is the square root of 10342656? Ans. 3216. 5. What is 964,5192360241 ? CASE 1.-To find a mean proportional between any two given numbers. RULE. Multiply the two given numbers together, and extract the square root of the product, which root will be the number sought. EXAMPLE. What is the mean proportional between 16 and 36? √36×16=24. Ans. CASE 2.-To find the side of a square equal in area to any given superficies. RULE.-Extract the square root of the given superficies, which root will be the side of the square sought. acre? EXAMPLES. 1. If an acre of land contains 160 square rods, what will be the side of a square, which should contain just an 160-12,649+rods. Ans. 2. A general having an army of 5184 men wishes to form them into a square; how many must he place in rank and file? 5184-72. Ans. 3. Let 8192 men be formed into an oblong, so that the number in rank may be double the file. 8192 64 in file. 64x2=128 in rank. 2 4. Suppose a gentleman would set out an orchard of 864 trees, so that the length shall be to the breadth as 3 to 2, and the distance of each tree, one from the other, 7 yards; how many trees must there be in length, and how many in breadth, and how many square yards of ground. do they stand on? To resolve any question of this nature, say, as the ratio in length is to the ratio in breadth, so is the number of trees to a fourth number, whose square root is the number in breadth; then as the ratio in breadth is to the ratio in length, so is the number of trees to a fourth number, whose root is the number in length. And as unity is to the distance, so is the number in length less by one to a fourth number; next do the same by the breadth, and multiply the two numbers thus found together, and the product will be the answer. As 3:2:864: 576,& ✓✓/576-24 num. in breadth. Ans. As2:3:864: 1296,& ✓1296=36 num in length Ans. As 1:7:36-1:245. And, as 1 : 7 :: 24—1 : 161. And 245×161=39445 square yards. Ans. CASE 3. To ascertain the proportionate capacities of water pipes. RULE.-Square the given diameter, and multiply said. square by the given proportion; the square root of the product is the answer. EXAMPLE. Admit 10hhds. of water are discharged through a leaden pipe of 2 inches diameter, in a certain time; what must be the diameter of another pipe, that shall discharge four times as much water in the same time ? 23=2,5 and 2,5×2,5=6,25 square. 4 given proportion. ✔25,00=5 inches diam. Ans. CASE 4.-The sum of any two numbers, and their product being given, to find each number. RULE. From the square of their sum, subtract 4 times their product, and extract the square root of the remain- |