The Root is a number whose continual multiplication into itself produces the power, and is denominated the square, cube, biquadrate, or 2d, 3d, 4th root, &c. accordingly, as it is, when raised to the 2d, 3d, 4th, &c. power, equal to that power. Thus, 4 is the square root of 16; because 4×4=16, and 3 is the cube root of 27, because 3×3×3=27; and so on. Although there is no number of which we cannot find any power exactly, yet there are many numbers, of which precise roots can never be determined. But by the help of decimals, we can approximate towards the root, to any assigned degree of exactness. The roots, which approximate, are called furd roots; and those which are perfectly accurate, are called rational roots. Roots are fometimes denoted by writing the character before the power, with the index of the root over it; thus the 3d root of 36 is expressed 3 ✓ 36, and the 2d root of 36 is ✓ 36, the index 2 being omitted when the square root is designed. If the power be expressed by several numbers, with the fign + or- between them, a line is drawn from the top of the fign over all the parts of it; thus, 3 the 3d roet of 47 + 22 is 47+32, and the ad root of 59-17 is 59-17, &c. Sometimes roots are designed like powers, with fractional indices; thus, the square root of 15 is 15, the cube root of 21 is 20 is 37--204, &c. 213, , and the fourth root of 37 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 so on, which points shew the number of figures the root will confift of. 2. Find 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 the square number, under the period, and subtract it therefrom, and to the remainder bring down the next period for a dividend. 3. Place the double of the root, already found, on the left hand of the dividend for a divisor. 4. Seek 4. Seek how often the divisor is contained in the dividend, (except the right hand figure) and place the answer in the root for the second figure of it, and likewife on the right hand of the divisor: Multiply the divisor, with the figure last annexed, by the figure last placed in the root, and fubtract the product from the dividend: To the remainder join the next period for a new dividend. 5. Double the figure already found in the root, for a new divifor, (or bring down your last divisor for a new one, doubling the right hand figure of it) and from these find the next figure in the root as last directed; and continue the operation in the fame manner, till you have brought down all the periods. Note 1. If, when the given power is pointed off as the power requires, the left hand period should be deficient, it must nevertheless stand as the first period. Note 2. If there be decimals in the given number, it must be pointed both ways from the place of units; If when there are in-. tegers, the first period in the decimals be deficient, it may be completed by annexing so many cyphers as the power requires; And the root must be made to confift of fo many whole numbers and decimals as there are periods belonging to each; and when the periods belonging to the given number are exhausted, the operation may be continued at pleasure by annexing cyphers. 2. Required the square root of 575,5? 575,50(23,98+root. 4 43)175 -469)4650 4221 4788)42900 38304 4596 remainder. 3. What is the square root of 10342656? Anf. 3216. 4. What is the square root of 964,5192360241? Anj. 31,05671. 5. What is the square root of Anf.,00563. Note. When more than half the root is found, the remaining figures of it may be found by Division, making use of the Jast divisor, and bringing down so many of the next figures of the refolvend, as there were periods to come down, when you began the division. RULES for the SQUARE ROOT of VULGAR FRACTIONS and MIXED NUMBERS. After reducing the fraction to its lowest terms, for this and all other roots; then, 1. Extract the root of the numerator for a new numerator, and the root of the denominator for a new denominator, which is the best method, provided the denominator be a complete power. But if it be not, 2. Multiply the numerator and denominator together; and the root of this product being made the numerator to the denominator of the given fraction, or made the denominator to the numerator of it, will form the fractional part required :-Or, 3. Reduce the vulgar fraction to a decimal, and ex tract its root. 4. Mixed numbers may either be reduced to im proper fractions, and extracted by the first or second rule; rule; or the vulgar fraction may be reduced to a decimal, then joined to the integer, and the root of the whole extracted. 81)81 Therefore 41=the root of the given fraction. 81 By Rule 2. 16X1681=26896 and 26896=164. Then, 1681-1644,09756+ 1681) 16(,0095181439+. And ✓,0095181439 By Rule 3. 09756+7 8208 2. What is the square root of ? 3. What is the square root of 424? Note. In extracting the square or cube root of any furd number there is always a remainder or fraction left, when the root is found: To find the value of which, the common method is to annex pairs of cyphers to the refolvend, for the square, and ternaries of cyphers to that of the cube, which makes it tedious to difcover the value of the remainder, pécially in the cube. Now this trouble may be faved by the following method. ef In the square the quotient is always doubled for a new divisor: Therefore when the work is completed, the root doubled is the true divisor, or denominator* to its own fraction; as, if the root be 12, the denominator * 1 Thefe denominators give a small matter too much in the square root, and too little in the cube, yet they will be fufficient in common und + |