8 3 315. 3 1 1 1 8 2 The power of any root may be obtained by multiplying the fractional exponent. Thus the 4th power of 273= 273. For by the last proposition V272 3/272 x 9 272 x 3/27%=3/278=273. The root of any power or root may be obtained by dividing the exponent by the index of the desired root. Thus 3/35 =35 This is the converse of the last proposition. For if the 3d power of 375 is 315, or 35, the 3d root of 35 must be 315. If the numerator and denominator of fractional indices be multiplied or divided by the same number, the value of the quantity is not altered. Thus, 36=312=33. For the multiplication of the numerator involves the number to a certain power, and the multiplication of the denominator extracts the corresponding root. Then the 3d root of the 3d power, the 5th root of the 5th power, &c., is the 1st power. We may multiply or divide any two roots of the same number, by adding or subtracting the fractional exponents. Thus, 12x v2=23 +1=28; 35= 75=51–1 = 51'a. For by the last proposition we have 2x V2=22* 923, which is equivalent to y25 or 28. Also 15-15 13/54 = 12,53='?/5 or 512. When the exact root of a number can be obtained, it is called a rational number. An irrational number, or surd, is one whose exact root cannot be obtained. Thus, V16, 327, \ 64, 781, are rational numbers, equivalent to 4, 3, 4, 3, respectively. . But 15, 19, V16, are all surds, and their roots can only be obtained approximately. , A number which has a rational root, is called a perfect power. Thus, 16 is a perfect 2d power, and a perfect 4th power, but an imperfect power of any other degree. Buť 5, 7, 12, &c., are imperfect powers of any degree. 1. What is the square root of 9? the cube root of 8? 1 2. What is the 4th root of 81 ? the 5th root of 32 ? 2 3. What is the value of vī; 251; V64; 641% ? 4. What is the product of vö by 912; V9 by nytt? 5. Multiply 1/3 by 9/3; 72 by 97; 43 by 42. 6. Divide 63 by 65 ; v5 by 55; 17'5 by 17. 7. Find the 4th power of võ; the 6th power of 88. 8. What is the cube root of 76; the 5th root of 114? 1 8, 9, EXTRACTION OF THE SQUARE ROOT. The square root of a number is the number which, when multiplied by itself, will produce the given number. In the following table are the numbers from 1 to 10 inclusive, and beneath them are their squares ; therefore, the numbers of the second line have for their square roots the numbers of the first. Roots 1, 2, 3, 4, 5, 6, 7, 10. Squares 1, 4, 9, 16, 25, 36, 49, 64, 81, 100. If there are decimals in the root, there will be twice as many in the square : because any product contains as many decimals as both factors. And conversely, there will be half as many decimals in the root as in the square. Every entire number, which is not the square of another entire number, is an imperfect second power. For the root of such a number cannot be expressed by a fraction, because a fraction multiplied by itself would give a fractional product,-it must therefore be a surd. The least square of units, is.... 1x1 = 1 The least square of tens is. 10X10 =100 The least square of hundreds, is.. 100 x 100=10000 The greatest square of units, is... 9x9 =81 The greatest square of tens and units, is. 99 x 99 =9801 The greatest square of hundreds, tens and units, is..... 999 x 999=998001 Hence we may see that if we divide a square number into periods of two figures, by placing a point over units, and one over each second figure to the left, the number of periods will denote the number of figures in the root. 37 Thus, the square root of i44 contains two figures, and is 12. The square root of 1600 contains 2 figures, and is 40. The square of 30 is 900. The square of 37 is 1369, 37 and to discover in what manner this square is formed, we will multiply 37 by 37, writing each product sepa 1369 (37 49 rately, instead of adding them as we proceed. We then 21 see that we must add the square of the tens, twice the 21 product of the tens by the units, and the square of the 9 units. 1369 Let us now reverse the process, and extract the square root of 1369. Pointing the periods, we find the root will consist of two figures, and the square of the tens 9 must therefore be contained in the 13 hundreds. The greatest square in 13 is 9, and the root 3 is 67) 469 written as the first figure of the required root. 469 Subtracting the square of the root already found, the remainder, 469, must contain twice the product of the tens by the units+the square of the units. To obtain the units' figure, we divide the 46 tens by 2x3=6 tens, which gives a quotient 7. Writing the 7 in the root, and also at the right of the divisor, we multiply by 7, and obtain 469, which is twice the product of the tens by the units plus the square of the units. Hence we deduce the following RULE. Separate the number into periods of two figures each, by placing a point over the units' figure, and another over each second figure to the left (and also to the right, if decimals are desired in the root). Write in the quotient the root of the greatest square contained in the left hand period, and subtract its square from the period. To the remainder annex the second period, and divide the tens of the number thus formed, by twice the first quotient figure, placing the result in the root, and also at the right of the new divisor. Multiply the completed divisor by the new quotient figure, subtract the product from the dividend, and annex the third period to the remainder. Double the root figures already found for a new trial divisor, and proceed as before, until all the periods are brought down. When any trial divisor is not contained in the tens of the dividend, place a zero in the root, and also at the right of the divisor, and bring down the next period. If any figure obtained for the root proves too large, diminish it by one and repeat the work. Approximate roots may be obtained by annexing decimal periods of two figures each. 1. Extract the square root of 529; of 961. 10. Find the approximate square root of 2, to six deci. mal places. 11. Find the approximate square root of 63; 1.6 ; .009; 27.1; 14367. 12. Find the square root of 318,829; 7}; 71&; 4; 173. 13. How long is the side of a square field that contains 225 acres ? 14. What is the side of a square which contains as much as a circle whose area is 273.5 square feet ? 15. An army of 8649 men is arranged in a solid square. How many are there on each side? 16. A certain oblong field contains 40 acres, and the length is 4 times the width. Required the length and breadth. The areas of circles are proportioned to the squares of their diameters. The areas of any similar figures are proportioned to the squares of their like dimensions. The area of any circle is equal to the square of its diameter multiplied by .7854. sec. sec. The circumference of a circle is equal to its diameter multiplied by 3.1416.* The area of a triangle is equal to the base multiplied by half the height. In any right-angled triangle, the square of the longest side is equal to the sum of the squares of the other two sides. The distance through which bodies fall, when falling freely, are as the squares of the times. In a vacuum, a body would fall 16,1 ft. in 1 second. Then we have the proportion, letting n represent any number of seconds. ft. ft. (1)2 : n2 :: 1612 : distance in n seconds. Any three terms of this proportion being given, the fourth may be readily found. But it should be remarked, that in consequence of the resistance of the air, the space actually fallen through is somewhat less than that given by the formula. 17. What is the diameter of a circle that is 16 times as large as one whose diameter is 13 feet. 18. The area of a circle is 7632 feet; what is the diameter? 19. A horse is fastened to a post in the centre of a field. What is the length of a rope that will allow hiin to graze an acre ? 20. The base of a triangle is 47 feet. What is the height, the area being 7691 square feet? 21. A ladder 75 feet long, rests against the limb of a tree that is 50 feet from the ground. How far is the foot of the ladder from the root of the tree ? 22. The length of a room is 18 feet, and the width 12 feet. What is the distance between the opposite corners ? What length of rope would reach from an upper corner to the opposite lower corner, the height being 10 feet? 23. The circumference of a circle is 29 rods. What is the side of a square having an equal area ? 24. Two ships left the same port; one sailed 125 miles * The more exact ratio is, 3.14159265358979323846264338328. |