2d power, 3x3=32, or 9; and the cube, or 3d power, 3×3×3=33, or 27; and so on. Thus the student will perceive, in finding the square of 3, there is only one multiplication, or two factors; in finding the cube there are two multiplications, or three factors, and so on. Involution is performed by the following RULE.-Multiply the given number, or first power, continually by itself, till the number of multiplications be one less than the index, or exponent of the power to be found, and the last product will be the power required. The powers of the nine digits, from the 1st power to the 5th, may be found in the following table 18 14 9 Biquadrates, or 4th Powers. |1|16| 81| 256 625|1296| 2401 4096 6561 or 3d Powers. 1 8 27] Sursolids, or 5th Powers. |1|32|243|1024|3125|7776 16807|32768|59049| 16 25 36 49 641 81 Cubes, 64 125 216 343 512 279 625-The 4th power or biquadrate. Answer. 3. What is the square or 2d power of 8? 4. What is the square of 40? Ans. 64. Ans. 1600. 5. What is the square of 500? Ans. 250,000. 6. What is the cube or 3d nower of 5? Ans. 125. NOTE.-A decimal fraction is involved or raised to any power, the same as a whole number, and the same rules are observed in pointing off as in Multiplication of Decimals A vulgar Fraction is raised to any power by multiplying the numerator of the fraction by itself, and the denominator by itself, till the number of multiplications be one less than the index, or exponent of the power to be found; then the power of the numerator, placed over the power of the denominator, gives the power of the fraction sought. If it be required to raise a mixed number to a certain power, first reduce it to an improper fraction, and then proceed as with a simple fraction. But those who desire it, may first reduce the given fraction to a decimal, and then raise the decimal to the power required. 10. What is the square or 2d power of ,5? 11. What is the cube, or 3d power of ,5? 12. What is the square of ? 13. What is the cube, or 3d power of ? 14. What is the square, or 2d power of 2? 15. What is the square of 44? Ans. ,25. Ans.,125. Ans. Ans. 343 Ans. 25. 64. Ans. 25-181 16. How much is 93, that is, the 3d power of 9? Ans. 729. Ans. 7776. Ans. 10,000. NOTE. It will be seen from the preceding examples, that raising a simple fraction, whether vulgar or decimal, to a higher power, diminishes it in the same proportion as a whole number becomes increased. EVOLUTION, Is the extracting or finding the roots of any given powers; or it is exactly the reverse of Involution. The root of any number or power is such a number, as being multiplied into itself a certain number of times, will produce that power.. Thus, 2 is the square root, or 2d reot of 4, because 22-2X2-4; and 3 is the cube root, or 3d root of 27, because 33=3×3×3=27. The power of any given number or root may be found exactly by multiplying the number continually into itself. But there are numbers, of which a proposed root can never be exactly found. Yet, by means of decimals, we may approximate or approach towards the root, to any degree of exactness. Those numbers whose roots only approximate towards the true root, are called surd numbers; but those whose roots can be exactly found, are called rational numbers. The Roots are sometimes denoted by writing the character be fore the power, with the index of the root against it. Thus, the square root of 25 is expressedV25, and the cube root of 64 is expressedv364; and the 5th root of 16807, 516807. The index to the square root is always omitted; the character only, being placed before it; thus, V16, the index 2, being omitted. When the power is expressed by several numbers, with the sign+, or; between, a line is drawn from the top of the sign over all the parts of it; thus the square root of 41-5, is√41-5, or thus, V(41-5,) enclosing the numbers in a parenthesis. But all roots are now frequently distinguished by fractional indices; thus, the square root of 8, is 8, the cube root of 64 is 64, and the square root of 41-5, is 41—5 (41—5)3. · or EXTRACTION OF THE SQUARE ROOT. Extracting the square root of any given number, is finding a number which, multiplied by itself, would produce the given number; consequently multiplying the root into itself, is a proof of the work. RULE.-Divide the given number into periods of two figures each, by setting a point over the place of units, another over the place of hundreds, and so on, over every second figure, both to the left hand in integers, and to the right hand in decimals. 4 Find the greatest square in the first period, on the left hand, and set its root, on 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 two figures of the next following period, for a dividend: Place twice the root, already found, on the left hand of the dividend for a divisor. Seek how often the divisor is contained in the dividend, (exclusive of the right hand figure,) and place the figure in the root, for the second figure of it, and likewise, on the right hand of the divisor; multiply the divisor, with the last figure annexed, by the last placed in the root, and subtract the product from the dividend; to the remainder, join the next period for a new dividend. 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. EXAMPLES. 1. What is the square root, or side of a square containing 36 square feet? Ans. 6 feet. S.. FIG. I. 6 feet. Operation. 36(6 Root. 36 6 36 Proof. DEM. The rule for the extraction of the square root, will appear obvious by attending to the process, by which any number is raised to the second power, or square. To find the second power of any number, we multiply the given number, or square root, by itself; therefore, to obtain the root from the power, we must inquire what number multiplied by itself, will produce the given number; we find on inquiring, that 6 multiplied by itself will produce the given number; therefore we place 6 in the quotient, for the root, and multiply it by itself, placing the product directly below the given number; we find that it equals the given number, consequently, 6 feet is the square root or side of a square, containing 36 square feet; which must appear plain to the student on inspecting Fig. I., because he will readily discover, that the figure contains 36 square feet, which is the 2d power of 6 feet, the square root or length of one of its sides. & feel. NOTE. There is a difference between square rods, square yards, and square feet, &c., and rods square, yards square, feet square, &c., thus: 1 ft. Three square feet may be represented by a Figure, 3 feet long, and one foot wide. And three feet square may be represented by a Figure 3 feet long, and 3 feet square are equal to 9 square feet. Such a Figure may be easily conceived of, by supposing two tiers of squares precisely like the one annexed, to be added to the bottom, which would give 9 equal squares. 3 feet. 2. A gentleman desirous of fencing into one square lot C25 square rods of land, wishes to know the length of one of the sides. 625(25 rods, the length of one side, or the square root of 625 square rods. 4 45)225 225 DEM. First, as our rule directs, we place a period () over the unit figure of our given number, and then one over the second figure beyond it. This we do, because any one figure multiplied by itself, will never produce more than two figures in the product. These periods also show, that our root will consist of as many figures as we have periods over the given number. We next seek the largest square in the left hand period, which we find to be 4, placing it under the first period, and writing its root, 2, in the quotient, for the first figure` of the root. We then subtract the 4, the square of the root already found, from 6, the first period; and to the right hand of the remainder, 2, we bring down 25, the next period, making 225 for our next dividend. Now it must appear obvious, on inspecting the annexed Figure, that, at all stages of the work, the quotient expresses the side of a square formed from what has been subtracted from the dividend; therefore our quotient, 2, must express the side of a square made up from what has been subtracted from our given number, 625; but the 2 in the quotient properly stands in the place of tens, because we must have another figure in the quotient, at the right hand of the 2, which gives it the local value of 20; consequently we may call the first figure of the root 20, which expresses one side of the square r s t u; therefore 20, the root, multiplied into itself, must give the number of square rods or area contained in the square r s t u already formed from what has been subtracted from the first period of our dividend. This will appear plain on inspecting the Figure r s t u, each side of which is 20 20X5=100 FIG. II. B A 5X5= 25 S 20 one side of square rstu 20 400 square rods in do. 100 do. do. A 8448 5-2 20X5=100 20 rods C rods, as expressed by the root already found; then the side of this square multiplied into itself, or the root already found, must give the number of square rods contained in the square r s t u, which has been subtracted, because 20 X 20=400, the number of square rods in the square rstu,which is just equal to the number subtracted from the dividend; for the 4 under 6, the first period, stands in the place of hundreds, consequently it expresses 400. Now it is evident, that 400 square rods of our dividend are disposed of, in forming the square rstu, each side of which is 20 rods; and we have now left of our dividend 225 square rods, which we must so dispose of, as to keep our Figure in a square form, after the additions are made, and also that our root may express one of the sides of the square after the Figure is completed. And in order to preserve our Figurer stu, in a square form, the additions must be made on two sides; therefore, as our rule directs, we double the root, and place it at the left hand of 225, our dividend, for an imperfect divisor; imperfect, because another figure must be placed B 100 do. do. do. in the 5-25 r. |