6 inches high; how many square yards of paper will it require to cover the walls? Ans. 252 square yards. Section 47. INVOLUTION. INVOLUTION is the raising of powers from any given number, as a root. A power is a quantity produced by multiplying any given number, called a root, a certain number of times continually by itself; thus, 3x3 ( 32. 3= 3 is the first power of 3=31. 9 is the second power of 3 3x3x3 27 is the third power of 3=33. 3x3x3x3 = 81 is the fourth power of 334. The number denoting the power is called the index, or exponent, of the power. Thus, the fifth power of 2 is 32, or 25; the third power of 4 is 64, or 43. To raise any number to any power required, we adopt the following RULE Multiply the given number continually by itself, till the number of multiplications be one less, than the index of the power to be found, and the last product will be the power required. Ans. 4096. Ans. 81. 1. What is the 3rd power of 5? 5×5×5=125 Ans. 2. What is the 6th power of 4? 3. What is the 4th power of 3? 4. What is the 1st power of 17? 5. What is the 0 power of 63 ? Ans. 17. Ans. 1. Section 48. EVOLUTION, OR THE EXTRACTION OF ROOTS. EVOLUTION, or the reverse of involution, is the extraction or finding the roots of any given power. The root is a number, whose continued multiplication into itself produces the power, and is denominated the square, cube, biquadrate, or second, third, fourth, &c., power, equal to that power. Thus, 4 is the square root of 16, because, 4 x4 = 16; and 3 is the cube root of 27, because, 3 × 3 × 3 =27; and so on. Roots, which approximate, are surd roots; and those, which are perfectly accurate, áre called rational roots. EXTRACTION OF THE SQUARE ROOT. 1. What is the square root of 625 ? To illustrate this question, we will suppose, that we have 625 tile, each of which is one foot square; we wish to know the side of a square room, whose floor they will pave or cover. If we find a number multiplied into itself, that will produce 625, that number will be the side of a square room, which will require 625 tiles to cover its floor. We perceive that our number (625) consists of three figures, therefore, there will be two figures in the root; for the product of any two numbers can have, at most, but just so many figures, as there are in both factors, and, at least, but one less. We will, therefore, for convenience, divide our number (625) into two parts, called periods, writing a point over the right hand figure of 625 (25 Ans. each period; thus, 625. We now find, that the greatest square number in the left hand period, 6 (hundred), is 4 (hundred); and that its root is 2, which we OPERATION. 400 45)225 225 write in the quotient (see operation). As this 2 is in the place of tens, its value must be 20 and its square 400. Let this be represented by a square, whose sides measure 20 feet each, and whose contents will, therefore, be 400 square feet. See figure 1. We now subtract 400 from 625, and there remains 225 square feet, to be arranged on two sides of figure 1, in order that its form may remain square. We therefore double the root 20, one of the sides, and it gives the length of the two sides to be enlarged; viz. 40. 20 FIG. I. 20 D 20 20 20 400 20 We then inquire, how many times 40, as a divisor, is contained in the dividend, and find it to be 5 times; this we write in the root, and also in the divisor. FIG. II. This 5 is the breadth of the addition to our square. (See figure 2.) And this breadth, multiplied by the length of the two additions (40) gives the contents of the two figures, E and F, 200 square feet, which is 100 feet for each. 20 20 E 20 G 5 5 100 25 D 20 20 20 5/20 There now remains the figure G, to complete the square, each side of which is 5 feet; it being equal to the breadth of the additions E and F. Therefore, if we square 5, we have the contents of the last addition, G = 25. It is on account of this last addition, that the last figure of the root is placed in the divisor. If we now multiply the divisor, 45, by the last figure in the root (5), the product will be 225, which is equal to the remaining feet, after we have formed our first square, and equal to the additions E, F, and G, in figure 2. We therefore perceive, that figure 2 may represent a floor 25 feet square, containing 625 square feet. From the above, we infer the following 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 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 integers, the first period in the decimals be deficient, it may be completed by annexing so many ciphers 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 numbers are exhausted, the operation may be continued at pleasure by annexing ciphers. NOTE 3. If it be required to extract the square root of a vulgar fraction, reduce the fraction to its lowest terms, then extract the square root of the numerator for a new numerator, and of the denominator for a new denominator; or, reduce the vulgar fraction to a decimal, and extract its root. 2. What is the square root of 148996 ? OPERATION. 148996 (386 9 68)589 766) 4596 Ans. 4879. 3. What is the square root of 23804641? Ans. 7856.4. 10. What is the square root of 9754.60423716 ? 12769 11. What is the square root of $72 ? 12. What is the square root of 1849 ? 13. What is the square root of ? 14. What is the square root of 129 ? 15. What is the square root of 60? 529 Ans. 98.7654. Ans.. Ans. 23. Ans. 1. Ans. 73. 16. What is the square root of 2857 ? Ans. 53. 17. What is the square root of 4717? Ans. 67. APPLICATION OF THE SQUARE ROOT. 18. A certain general has an army of 226576 men; how many must he place rank and file to form them into a square? Ans. 476. NOTE. In a right angle triangle, the square of the longest side is equal to the sum of the squares of the other two sides. 19. What must be the length of a ladder to reach to the top of a house 40 feet in height; the bottom of the ladder being placed 9 feet from the cill? Ans. 41 feet. 20. Two vessels sail from the same port; one sails due north 360 miles, and the other due east 450 miles; what is their distance from each other? Ans. 576.2 miles. |