By examining the several powers of 3 in the examples given, we see that the index of each power is equal to the number of times 3 is used as a factor in the multiplications producing the power, and that the number of times the number is multiplied into itself is one less than the power denoted by the index. Hence the RULE. Multiply the given number continually by itself, till the number of multiplications is one less than the index of the power to be found. and the last product will be the power required. NOTE 1.-- A fraction may be raised to any power by this rule, by mul tiplying its terms continually together. Thus, the second power of is NOTE 2.- - A mixed number may be either reduced to an improper fraction, or the fractional part reduced to a decimal, and then raised to the required power. EXAMPLES FOR PRACTICE. 1. What is the 2d power of 6? ART. 265. To raise a number to any required power without producing all the intermediate powers. Ex. 1. What is the 8th power of 4? Ans. 65536. 64 64 16 65536. We first raise 4 to the 3d power, and write the exponents denoting each power directly above it. We then add the exponent 3 to itself, and increasing the sum by the exponent 2, obtain 8, a number equal to the power required. We next multiply 64, the power belonging to the exponent 3, into itself, and this product by 16, the power belonging to the exponent 2, and obtain 65536, for the 8th power. Hence the following RULE. - Raise the given number to any convenient power, and write the exponents denoting the respective powers directly above them. Then QUESTIONS. What is the rule for raising a number to any required power? How may a vulgar fraction be raised to a required power? How a mixed number? Art. 265. What are the numbers placed over the several powers of 4 called, and what do they denote? add together such exponents as will make a number equal to the required power, repeating any one when it is more convenient; and the product of the powers belonging to these exponents will be the required answer. NOTE. - When a power has been found, double that power may be ob tained by multiplying it into itself. EXAMPLES FOR PRACTICE. 2. What is the 7th power of 5? 3. What is the 9th power of 6? 4. What is the 12th power of 7? 5. What is the 8th power of 8? 6. What is the 20th power of 4? 7. What is the 30th power of 3? Ans. 78125. Ans. 10077696. Ans. 13841287201. Ans. 16777216. Ans. 1099511627776. Ans. 205891132094649. Ans. 1125899906842624. х § XXXIX. EVOLUTION. ART. 266. EVOLUTION is the method of finding the root of a given power or number, and is therefore the reverse of Involution. A root of any power is a number which, being multiplied into itself a certain number of times, produces the given power. Thus, 4 is the second or square root of 16, because 4 X 4: 16; and 3 is the third or cube root of 27, because 3 × 3 × 3 27. The root takes the name of the power of which it is the root. Thus, if the number is a second power, the root is called the second or square root; if it is a third power, the root is called the third or cube root; and, if it is a fourth power, its root is called the fourth or biquadrate root. Those roots which can be exactly found are called rational roots; those which cannot be exactly found, but approximate towards the true root, are called surd roots. QUESTIONS. -What is the rule for involving a number without producing all the intermediate powers? Art. 266. What is Evolution? What is a root? From what does the root take its name? What are rational roots? What surd roots? Roots are denoted by writing the character ✔ before the power, with the index of the root over it, or by a fractional index or exponent. The third or cube root of 27 is expressed thus, 27 or 27; and the second or square root of 25 is expressed thus, 25 or 254. NOTE.-The index 2 over is usually omitted, when the square root is required. Thus, 64 denotes the square root of 64. EXTRACTION OF THE SQUARE ROOT. ART. 267. The Square Root is the root of any second power, and is so called because the square or second power of any number represents the contents of a square surface, of which the root is the length of one side. ART. 268. To extract the square root of any number is to find a number which, being multiplied by itself, will produce the given number. The following numbers in the upper line represent roots, and those in the lower line their second powers or squares. 1 2 3 4 5 6 7 8 9 10 Roots, 16 25 36 49 64 81 100 It will be observed that the second power or square of each of the numbers above contains twice as many figures as the root, or twice as many wanting one. Hence, To ascertain the number of figures in the square root of any given number, it must be divided into periods, beginning at the right, each of which, excepting the last, must always contain two figures; and the number of periods will denote the number of figures of which the root will consist. Ex. 1. I wish to arrange 625 tiles, each of which is 1 foot square, into a square pavement; what will be the length of one of the sides? Ans. 25 feet. OPERATION. 625(25, Ans. 4 45)225 225 It is evident, if we extract the square root of 625, we shall obtain one side of the pavement, in feet. (Art. 267.) Beginning at the right hand, we divide the number into periods, by placing a point over the right hand figure of each period; and then find the greatest square number in the QUESTIONS. - How are roots denoted? How is the third or cube root denoted? How the second or square root? What is said of the index 2? -Art. 267. What is meant by the square root, and why is it so called? Art. 268. What is meant by extracting the square root? How many more figures in the square of any number than in the root? How do you ascertain the number of figures in the square root of any number? Why do you point off the numbers into periods of two figures each? What is found by extracting the square root of the number in the question? left hand period, 6 (hundreds) to be 4 (hundreds), and that its root is 2, which we write in the quotient. As this 2 is in the place of tens, its value is 20, and represents the side of a square, the area or superficial contents of which are 400 square feet, as seen in Fig. 1. 20 feet. Fig. 1. 20 feet. D 20 20 400 20 feet. 20 feet. We now subtract 400 feet from 625 feet, and have 225 feet remaining, which must be added on two sides of Fig. 1, in order that it may remain a square. We therefore double the root 2 (tens) or 20, one side of the square, to obtain the length of the two sides to be enlarged, making 40 feet; and then inquire how many times 40, as a divisor, is contained in the dividend 225, and find it to be 5 times. This 5 we write in the quotient or root, and also on the right of the divisor, and it represents the width of the additions to the square, as seen in Fig. 2. The width of the additions being multiplied by 40, the length of the two additions, makes 200 square feet, the contents of the two addi 25 feet. Fig. 2. 25 feet. G 25 E 20 5 100 D 20 20 גי 20 5 tions E and F, which are 100 feet for each. The space G now remains to be filled, to complete the square, each side of which is 5 feet, or equal to the width of E and F. If, therefore, we square 5, we have the contents of the last addition, G, equal to 25 square feet. It is on account of this last addition that the last figure of the root is placed in the divisor, for we thus obtain 45 feet for the length of all the additions made, which, being multiplied by the width, (5ft.,) the last figure in the root, the product, 225 25 feet. square feet, will be the contents of the three additions E, F and G, and equal to the feet remaining after we had found the first square. Hence we obtain 25 feet for the length of one side of the pavement, since 25 X 25625, the number of tiles to be arranged, and equal to the sum of the several parts of Fig. 2; thus, 400 +100+100+25 625. From this solution and explanation we deduce the following QUESTIONS. What is first done after dividing the number into periods? What part of Fig. 1 does this greatest square number represent? What place does the figure of the root occupy, and what part of the figure does it represent? Why does it have the place of tens? Why do you double the root for a divisor? What part of Fig. 2 does the divisor represent? What part does the ast figure of the root represent? Why do you multiply the divisor by the last figure of the root? What parts of the figure does the product represent? Why do you square the last figure of the root? What part of the figure does is square represent? What other way of finding the contents of the additions without multiplying the parts separately by the width? RULE.1. Separate 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, and these points will show the number of figures of which the root will consist. 2. Find the greatest square number in the first or left hand period, placing 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, subtracting 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. Find how often the divisor is contained in the dividend, omitting 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 the 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 all the periods have been brought down. NOTE.-1. If the dividend does not contain the divisor, a cipher must be placed in the root, and also at the right of the divisor; then, after bringing down the next period, this last divisor must be used as the divisor of the new dividend. 2. When there is a remainder after extracting the root of a number, periods of ciphers may be annexed, and the figures of the root thus obtained will be decimals. 3. If the given number is a decimal, or a whole number and a decimal, the root is extracted in the same manner as in whole numbers, except in pointing off the decimals either alone or in connection with the whole number, we begin at the separatrix and place a point over every second figure toward the right, filling the last period, if incomplete, with a cipher. 4. The square root of any number ending with 2, 3, 7, or 8, cannot be exactly found. EXAMPLES FOR PRACTICE. 2. What is the square root of 148996 ? *The figure of the root must generally be diminished by one or two units, on account of the deficiency in enlarging the square. QUESTIONS.-What is the rule for extracting the square root? What is to be done if the dividend does not contain the divisor? What must be done if there is a remainder after extracting the root? What do you do if the given number is a decimal? Of what numbers can the square root not be found? |