9th power. INVOLUTION, OR THE RAISING OF POWERS. 147 The number which denotes a power is called its index. Note. When any power of a vulgar fraction is required, first raise the numerator to the required power and then the denominator to the required power, and place the numerator over the denominator as before, 2 × 2 × 2 × 2. thus, the 4th power of 3x3x3x3 Table of the first nine powers. 81 512 6564 19683 16384 65536 262144 1953125 39 27 81 243 629 2187 Questions. What is the product arising from the multiplication of any figure by itself a given number of times called? What is the number which denotes a power called? How do you proceed to find any required power of a vulgar fraction? Examples. 1. What is the square of 32 ? 32 64 96 2. What is the cube of 14? 3. What is the sixth power of 2.8? Ans. 2744. Ans. 481.890304. 4. What is the third power of .263 ? Ans. .018191447. 5. What is the eighth power of ? Ans. 356. EVOLUTION, OR THE EXTRACTING OF The root of a number is that which will produce that number by being multiplied by itself a given number of times. The object proposed by the extraction of the root of a number is to find that number which being multiplied by itself a number of times equal to that for which the root is required will produce that number. SQUARE ROOT. When the square root of any given number is required. 1. Separate the given number into periods of two figures each beginning at the right hand or units place. Note.-If the square root of a whole number and decimal is required point the whole numbers as before, and then commence at the decimal point and count periods of two figures each towards the right, observing if there is only one figure at the last to place a cipher to its right to make an even period. When a decimal only is given, separate the periods in the same way. 2. Find the greatest root of the first left hand period and place it to the right of the given sum and its square under said period and take their difference. 3. Bring down the next period and set it to the right of the remainder, as in long division for a dividend. 4. After bringing down the period, double the ascertained root, and place it to the left of the remainder for a divisor. 5. Try how often the divisor is contained in the dividend, omitting the last figure, and place the re. sult to the right of the ascertained root, and to the right of the number produced by doubling the ascertained root. 6. Multiply as in long division, and proceed with the operation until all the periods have been brought down. Note. When the square root of a fraction is required, extract the square root of the numerator, for a new numerator, and square root of the denominator for a new denominator. If there be a remainder either to the numerator or denominator, reduce the fraction to a decimal and extract the square root. Questions. When the square root of any given number is required, how do you prepare the given sum? What is to be noted when whole numbers and decimals, and decimals only are given? After separating the given number into periods of two figures, what is to be done? After having found the greatest root of the first left hand period, and placed it to the right of the given sum and its square under the first period, and taken their difference what is to be done? When you have brought down the next period and placed it to the right of the remainder for a dividend, how do you proceed? When you have doubled the ascertained root, and placed it to the right of the divisor, what is to be done? When you have found how often double the ascertained root is contained in the dividend with one figure omitted, what is next to be done, and how do you proceed till the operation is completed? What is to be noted when the square root of a vulgar fraction is required? Examples. 1. What is the square root of 6.9169? 6.9169(2.63 Ans. 4 46)291 276 523) 1569 2. What is the square root of 39375655? Ans. 6275.+ 3. What is the square root of 1486.17901? Ans. 38.55.+ 4. What is the square root of 96385163? Ans. 9817.+ 5. What is the square root of .000132496 ? 6. What is the square root of 18.362147? 3200 1764 500 Ans. 01151.+ Ans. 4.285.+ Ans. 7. Ans. . Ans. 8. Ans. 7. Ans. 5. 7. What is the square root of 2450? 8. What is the square root of 1294 9. What is the square root of 320 ? 10. What is the square root of 50% 11. What is the square root of 30,5% ? 12. An employer paid 1296 dollars to a number of men, and each man received as many dollars as here were men, how many men were there? Ans. 36 men. 100 13. Supposing a square tower had on each side an equal number of windows, and the whole number was 169, how many were there on one of its sides? Ans. 13. 14. A certain square piece of land contains 3097600 square yards, the length of one of its sides is required. Ans. 1 mile. Note. The square of the longest side of the right angled triangle is equal to the sum of the squares of the other two sides, and consequently the difference of the square of the longest, and either of the others, is the square of the remaining one? 15. If the height of a fort be 15 feet, and surrounded by a ditch 24 feet wide, what must be the length of a ladder to reach from the outside of the ditch to the top of the fort? Ans. 28.3+ feet. 16. What is the height of a castle, when a line 212 feet long will just reach from the top of the castle to the opposite bank of a river, known to be 20 yards broad? Ans. 203.332 feet. CUBE ROOT. When the cube root of any number is required. 1. Prepare the given number by separating it into periods of three figures each from the units place. Note. When whole numbers and decimals; or decimals only, are given the same observation is to be made as to the manner of separating the figures into periods as in the square root. 2. Find the greatest root contained in the left hand period, place it to the right of the given number and its cube under the first left hand period, and take their difference, bring down the next period, and place it to the right of said difference for a dividend. 3. Square the root and multiply the square by three, for a defective divisor. 4. Try how often the defective divisor is contained in the dividend, omitting the two right hand figures, and place the number of times it is contained to the right of the defective divisor, supplying the place of tens with a cipher if the square be less than 10. 5. Multiply the last figure by all the figures of the root previously ascertained, and multiply that product by 30, then add the product to the divisor to complete it. Multiply and subtract as in long division and bring down the next period for a new dividend continually, until all the periods have been brought down. |