318 



THE POPULAR EDUCATOR. 



LESSONS IN ARITHMETIC. XIX. 



SQUARE AND CUBE EOOT (continued). 



9. THE square root of a fraction is obtained by taking the 

 equare root of the numerator for a numerator, and the squaro 

 root of the denominator for a denominator. This follows at once 

 from the consideratio* that the multiplication of fractions is 

 effected by multiplying the numerators for a numerator, and the 

 denominators for a denominator. When either the numerator 

 or the denominator is not a complete square, in which case the 

 fraction, itself evidently has no exact square root, instead of 

 finding an approximate root of both numerator and denominator 

 in decimals, and then dividing one by the other, it will bo better 

 first to reduce the fraction to a decimal, and then to take the 

 square root. 



EXAMPLE. To find the square root of f. 



Reducing f to a decimal, we find it to be '285714 (see Lesson 

 XVI., Art. 21). 



Hence we should find by the previous method the square root 



of '28571428571428 ... to as many decimal places as we please, 

 by continually taking in more and more figures of the recurring 

 periods. 



Similarly, in finding the square root of f , we should proceed 



thus : = '4, and then find the squaro root of '400000, etc., to 

 as many places as we please. 



06s. It does not follow that because the numerator and 

 denominator of a fraction are not complete squares, that the 

 fraction has no square root ; for the division of numerator and 

 denominator by some common measure may reduce them to 

 perfect squares. Thus, ||, when numerator and denominator are 

 divided by 7, gives J, the square root of which is f. A fraction 

 must bo reduced to its lowest terms to determine whether it be 

 a complete square or not. 



10. Abbreviated Process of Extraction of Square Root. 



When the squaro root of a number is required to a consider- 

 able number of decimal places, we may shorten the process by 

 the following 



Rule for tJie Contraction of the Square Root Process. 



Find by the ordinary method one more than half the number 

 of figures required, and then, using the last obtained divisor as 

 a divisor, continue the operation as in ordinary long division. 



EXAMPLE. Find the square root of 2 to 12 figures. 



2-0000, etc. ( 1-414213 | 56237 



24) 100 



281 ) 400 



281 



2824) 11900 

 11296 



28282 ) 60400 

 56564 



282841 ) 383600 

 282841 



2828423) 10075900 

 8485269 



15906310 

 14142115 



17641950 

 16970538 



6714120 

 5656846 



10572740 

 8485269 



20874710 

 19798961 



1075749 

 Here, having obtained by the ordinary process the first seven 



figures, we get tho rest by dividing as in ordinary division by 

 the last divisor, 2828423. 



11. We might extract the square root of a perfect square by 

 splitting it into its prime factors, but unless the number is not 

 large this would be a tedious method. 



EXAMPLE. Find the square root of 441. 



Following the method given in Lesson VIII., Art. 5 

 3)441 



3)147 



Therefore 441 = 3 2 X 7 2 ; of which the square root is 3 X 7, 

 or 21. 



06s. Unless a number is made up of prime factors, each of 

 which is repeated an even number of times, it is not a perfect 

 square. 



EXERCISE 39. 



1. Find the square root of the following numbers : 



1. 529. 



2. 5329. 



3. 784. 

 '4. 4761. 



5. 7056. 



6. 9801. 



7. 27889. 



8. 961. 



9. 97 to 4 places of decimals. 

 10. 190 to 5 places. 



11. -81796 to 4 places. 



12. 1169 64. 



13. 3-172181 to 4 places. 



14. 10342G56. 



15. 2li lint ?S> -?Vi s ;- 



16. -| to 4 places. 



17. 17-J to 4 places. 



18. 964-51923G02U. 



19. -00000025. 



2. Find the square root of the following numbers by the 

 abbreviated method : 



1. 365 to 11 figures ill the root. 3. 3 to 17 figures. 



2. 2 to 12 figures. 



3. Extract tho square root of 2116, 21316, and 7056, by 

 splitting them into their prime factors. 



12. Extraction of the Cube Root. 



To extract the cube root of a given number is the same thing 

 as resolving it into three equal factors. 



As in the case of the square root, we must content ourselves 

 with giving, without explanation of the reason of its truth, the 



Rule for the Extraction of the Cube Root of a given number. 



Mark off the given number into periods of three figures each, 

 by placing a point over the figure in the unit's place, and then 

 over every third figure to the left (and to the right also, if there 

 be any decimals). Put down for the first figure of the root the 

 figure whose cube is the greatest cube in the first period, and 

 subtract its cube from the first period, bringing down the next 

 period to the right of the remainder, and thus forming a number 

 which we shall call a dividend. Multiply the squaro of the part 

 of the root already obtained by 3 to form a divisor, and then, 

 having determined how many times this divisor is contained in 

 the dividend without its two right-hand figures, annex this 

 quotient to tho part of the root already obtained.* Then deter- 

 mine three lines of figures by the following processes : 



1. Cube tbe last figure in the root. 



2. Multiply all tbe figures of the root except the last by 3, and the 

 result by the square of the Just. 



3. Multiply the divisor by the last figure in the root. 



Set down these lines in order, under each other, advancing 

 each successively one place to the left. Add them up, and 

 subtract their sum from the dividend. Bring down the next 

 period to tho right of the remainder, to form a new dividend, 

 and then proceed to form a divisor, and to find another figure of 

 the root by exactly the same process, continuing the operation 

 until all the periods are exhausted. 



13. In decimals, the number of decimal places in tho cubs 

 root will be the same as the number of points placed over the 

 decimal part, i.e., as the number of periods in the decimal part. 



05s. If, finally, there be a remainder, then tho given number 

 has no exact cube root, but, as in tho case of the squaro root, an 

 approximation can be carried to any degree of nearness by 

 adding ciphers, and finding any number of decimal places. 



The rule will be best understood by following the stops of an 

 example. 



* It will be found necessary sometimes, as will be seea by the 

 exnmple given in Art. 15, to set down as the next figure in the root, 



one less than this quotient. 



