IN 



LKSSn.NS IN AKlTH.MKTir.-XVIll. 



SgUAKK AM) CUIJE BOOT. 



I already stated that when any number in multiplied 

 t any number of time*, th pMduot* are called the second, 

 third, fourth powers, etc., of tho nuinbor respectively. 



The second and lliird powen of any number are generally 

 called the square and cube of that number. Tliun, 81 U tho 

 square of 'J, 27 IM tho cube of 3. 



Any power of a number is expressed by writing the number of 

 the power in small figures aboro th> littlo to the right. 



Thin, the i would be written 1 J ; the cube o; ' 



th- tit'th power of 7, 7 f ; and so on. 



ersely, tho nuinbor which, taken twice an a factor, will 



it |_'i\.n ininil.er, i-t called the square root of that num- 



at which, taken three times as a factor, will produce a 



'ailed the cube root of**it ; that which, taken 



four tiinei as a factor, will produce a given number, is called 



"ftk root of it ; and BO .on. 



Any root of a number is represented by writing the sign */ 

 over tho number, and placing tho number corresponding to the 

 number of tho root on the loft of the symbol, thus : V$ indi- 

 cates tho cube root of 8, \/8l tho fourth root of 81. 



The square root of a number is generally expressed by merely 

 writing the symbol V over tho number, without the figure 2. 

 Thus, K/3 moons the square root of 3 ; V84 the square root 

 of 84. 



2. Every number has manifestly a 2nd, 3rd, 4th, etc., 

 But every number has not conversely an exact square, cube, 

 third root, etc. For example, there is no whole number which, 

 when multiplied into itself, will produce 7 ; and since any frac- 

 tion in its lowest terms multiplied into itself must produce a 

 fraction, 7 cannot have a fraction for its square root. Hence 7 

 has no ernct square root. But although wo cannot find a whole 

 number or fraction which, when multiplied into itself, will pro- 

 duce 7 exactly, we can always, as will be shown hereafter, find 

 a decimal which will be a very near approximation to a square 

 root of 7, and wo can carry the approximation as nearly to n/7 

 as we please. And the same will be true of every number which 

 has no exact square root, third root, etc. 



It is desirable that tho student should know by heart tho 

 squares and cubes of the successive numbers from 1 up to 12, 

 appended in tho following table : 



In finding tho square of any number which is not very largo 

 under 100, say tho following method will be found useful : 



3. SJwrt Method for finding the Square of a Number. 



Add and subtract from tho number its defect or excess from 

 the nearest multiple of 10. Multiply the numbers BO found 

 together, and add the square of the defect or excess. 



For instance, to find the square of 97 : 



100 is the nearest multiple of 10, and 3 is the defect of 97 from it. 

 97 + 3-100 

 97-3-94 

 3' - 9. 

 'ore the required square of 97 is 100 x 94 + 9 9409. 



Again, to square 44 : 



40 is the nearest multiple of 10 to 44, and 4 is the excess of 44 



over it. 



44 + 4 48 



44-4-40 



4 - 16. 



Hence the required square is 1920 + 16, or 1936. 



This operation con be readily performed mentally, as will bo 

 found by a little practice. 



4. Observe, also, that no square number con end in 2, 3, 7, or 

 8 ; but that a cube can terminate in any one of the 10 figures. 



A number which ha* an exact square root U 

 perfect square. 



EXKUCIHK 38. 



(1.) Square the following number* bj the method of Art Si 

 >7, 45, 68, 79, V- 



! 'otermine whether the following number* are parfatt 

 squares or perfect cubeti ; and where they are not, find tit* lea*t 

 multiplier which will make them *o : 72, 125, 164, 1355, 43*4, 

 5010, 4096. 



(.'{.) Take any two number*, and prove that th difference of 

 their square* i* equal to the product of their sum and itilfnune*. 



(4.) Take any two number*, and prove that the Jif*M*m of 

 their cube* divided by their difference i* equal to the sum of 

 their squares and their product. 



(5.) Take any two number*, and prove that their product i* 

 equal to the square of half their *nm the square of half their 

 difference. 



5. Extraction of the Square Root. 



The square root of any given whole number or decimal can be 

 obtained, or extracted, as is sometimes said, by means of the 

 following rule, which we give without proof, as it require* the 

 aid of algebra to establish it satisfactorily: 



Rule for the Extraction >/ the Square Root of any number. 



Separate tho given number into periods containing two figure* 

 each, by placing a point over the unit's figure, and also over 

 every second figure towards the left in whole numbers, but both 

 towards the left and the right in decimals. 



Subtract from the extreme left-hand period the greatest 

 square which is contained in it, and put down it* square root 

 for the first figure of tho required whole square root To the 

 right of the remainder bring down the next period for a 

 dividend. Double tho part of the square root already found, 

 and place it on the left of this dividend for a partial divisor ; 

 find how many times it is contained in the dividend, omitting 

 its right-hand figure, and annex this quotient to the part of the 

 root already obtained, and also to the partial divisor. Multiply 

 the divisor thus formed by the last figure of the root, and sub- 

 tract the product from the dividend, bringing down the next 

 period to tho right of the remainder for a dividend. Continue 

 the operation until all the periods have been brought down. If 

 tho original number bo a decimal, the process above indicated 

 must be performed as if it were a whole number, and- a number 

 of decimal places cut off from the root so obtained, equal to the 

 number of points placed over the decimal part of the original 

 number. 



6. The process will be best followed by means of examples. 

 EXAMPLE 1. Find the square root of 627264. 



The greatest square in tho first period 62 is the square of 7 or 

 49. Subtracting 49 from 62, we place 7 as the 

 first figure of the root. We bring down the 

 next period 72 to the right of the remainder 13, 

 for a dividend, doubling 7 to form a partial 

 divisor, which is contained in 137 (tho dividend 

 without the right-hand figure 2) 9 times. We 

 annex the 9 both to the partial divisor and to 

 the part of the root already obtained. Multi- 

 plying 149 by 9, we subtract the product 1341 

 from the dividend, and bring down the next .... 



period, 64, to the right of the remainder for a 

 dividend, doubling 79, the part of the root already obtained, for 

 a partial divisor. 158 is contained 2 times in 316, and annexing 

 the 2 both to the partial divisor 158 and to 79, the part of the 

 root already obtained, we multiply thr divisor 

 7-34-li (271 1582 by this lost figure of the root ; the product 

 4 is 3164, which, subtracted from the dividend, 



leaves no remainder. Hence 792 is the exact 

 square root of 627264. 



EXAMPLE 2. Find tho square root of 7 

 Placing a dot over the figure in the unit's 

 place, we put one over every second figure to 

 the right, and then, performing the operation as 

 if 73441 were a whole number, as indicated in 

 the margin, wo pet 271 as the root We cnt 

 off two decimal places from this, because there are two dote 

 over the decimal part of the original decimal. 

 The square root is therefore 2 "71. 

 06s. At any stage of the process, the product of tho com- 



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us i ;rj 



PJM 



3161 



47)334 

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541) 



541 

 541 



