EVOLUTION, OR THE EXTRACTION OF ROOTS. 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, &c. power, equal to that power. Thus, 4 is the square root of 16, because 4x4-16, 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 ap. proximate toward 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. Roots are sometimes denoted by writing the character √ before the power, with the index of the power over it; thus the 3d root 3 of 36 is expressed v 36, and the 2d root of 36 is ✔ 36, the index 2 being omitted when the square root is designed. or If the power be expressed by several numbers, with the sign+ between them, a line is drawn from the top of the sign over 3. all the parts of it. Thus the 3d root of 47+22 is √47+22, and 17 is v59-17, &c. the 2d root of 59 Sometimes roots are designated like powers, with fractional indices. Thus, the square root of 15 is 152, the cube root of 21 is 21 and 4th root of 37 -20 is 37-201 &c. The denominator shows the root which is to be extracted, and the numerator shows the power to which that root is to be raised. Or the number may be raised to the power indicated by the numerator, and the root, indicated by the denominator, then extracted. Thus 64*=42=16, 3 2 3 =√64 =√ 4096=16. Hence the square of the cube root of any quantity is the same as the cube root of the square of the same quantity. The index or exponent of the root is one more than the number of multiplications, required to produce the given number or power. Square Cubes Sqd. or 12th. Pow. 1 4096 531441 16777216 244140625 2176782336 13841287201 68719476736 282429536481 Fourth Sursolids, for 13th. Pow. 1 8192 1594323 67108864 1220703125 13060694016) 96889010407 549755813888 2541865828329 2d. Sursolids Sqd. or 14th. Pow. 1 16384 4782969 268435456 6103515625 78364164096 678223072849 4398046511104 22876792454961 Sursolids Cubed, or 15th. Pow. 1 32768 14348907 1073741824 30517578125 470184984576 4747561509943 35184372088832 205891132094649 9765625 60466176 282475249 1073741824 3486784401 177147 4194304 48828125 362797056 1977326743 8589934592 31381059609 THE EXTRACTION OF THE SQUARE ROOT. 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, which points show the number of figures the root will consist 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 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 likewise on the right hand of the divisor: Multiply the divisor with the figure last annexed by the figure last placed in the root, and subtract the product from the dividend: To the remainder join the next period for a new dividend. 5. Double the figures already found in the root, for a new divisor, (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 same 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 figure 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 order to show the reason of the rule, it will be proper to premise the following Lemma. The product of any two numbers can have, at most, but so many places of figures, as are in both the factors, and at least but one less. Demonstration. Take two numbers consiting of any number of places; but let them be the least possible of those places, viz. Unity with cyphers, as 100 and 10: Then their product will be 1 with so many cyphers annexed as are in both the numbers, viz. 1000; but 1000 has one place less than 100 and 10 together have: And since 100 and 10 were taken the least possible, the product of any other two numbers, of the same number of places, will be greater than 1000: consequently, the product of any two numbers can have, at least, but one place less than both the factors. Again, take two numbers, of any number of places, which shall be the greatest possible of those places, as 99 and 9. Now, 99 X9 is less than 99 × 10; but 99 × 10 (=990) contains only so many places of figures as are in 99 and 9; therefore, 99 × 9, or the product of any other two numbers, consisting of the same number of places, cannot have more places of figures, than are in both its factors. Corollary 1. A square number cannot have more places of figures than double the places of the root, and at least but one less. Corollary 2. A cube number cannot have more places of figures than triple the places of the root, and at least but two less. integers, 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 consist of so 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 ciphers. EXAMPLES. 1st. Required the square root of 30138696025? 30138696025(173605 the root. 1 1st. Divisor=27)201 189 2d. Divisor-343)1238 1029 3d. Divisor 3466)20969 20796 4th. Divisor-347205)1736025 1736025* 2d. Required the square root of 575.5? 575-50(23-98+, the root. 4 43)175. 129 469)4650 4788)42900 4596 Remainder. *The Rule for the extraction of the square root, may be illustrated by attending to the process by which any number is raised to the square. The several products of the multiplication are to be kept separate, as in the proof of the rule for Simple Multiplication. Let 37 be the number to be raised to the square. 37X37=1369: = 37X37 3d. What is the square root of 10342656? 5th. What is the square root of 234.09? 6th. What is the square root of 0000316969? 7th. What is the square root of 045369? 4th. What is the square root of 964-5192360241? Ans. 31-05671, Ans. 3216. Ans. 15.3. Ans. 00563 Ans. 213. RULES For the Square Root of Vulgar Fractions and Mixed Numbers. After reducing the fraction to its lowest terms, for this and all others roots; then, 1st. 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, 2d. Multiply the numerator and denominator together; and the root of this product being made the numerator to the denominator of the given factor, or made the denominator to the numerator of it, will form the fractional part required.* Or, Now, it is evident that 9, in the place of hundredths, is the greatest square in this product; put its root, 3, in the quotient, and 900 is taken from the product. The next products are 21+21=2x3x7, for a dividend. Double the root already found, and it is 2x3, for a divisor, which gives 7 for the quotient, which annexed to the divisor, and the whole then multiplied by it, gives 2×3×7(=42)+7X7(=49) which placed in their proper places, completely exhausts the remainder of the square. The same may be shown in any other case, and the rule becomes obvious. Perhaps the following may be considered more simple and plain. Let 37,= 30+7, be multiplied, as in the demonstration of simple multiplication, and the products kept separate. 30+7 30+7 V7X2 The reason of which is, that the value of a fraction is not altered by multiplying both its parts by the same quantity. Thus V1= But VXV2= |