IM 



MATHEMATICS. ALGEBRA. [EXTRACTION OF THE CDBB HOOT. 



bered : the following is an example of its applica- 

 tion ^ 



Convert the fraction .1. . . into an equivalent 



one with a rational denominator. 



6 80 



V6 V 2 " 8 8 



And this form, though involving Vine cube roots, is 

 of easier calculation than the original, if tables of cubo 

 root* are used; because there is no long-division operation. 



EXAMPLES FOE EXERCISE. 



Prove that 



1 



, 



- 



2+.V 3 3 + 



. 

 3 + V 3 6 



3. -,- 



To Extract the Cube Root of a Compound Quantity. 



In order to discover tho means of arriving at the cube 

 root of a polynomial, we may proceed in imitation of 

 the course adopted for the determination of the square 

 root. Thus, to begin with the simplest case, let its take 

 the cube of a -f- 6 ; that is, the expression a 3 -j- 3a 2 b -f- 

 Zab 3 -\- 6 3 ; and, availing ourselves of our previous know- 

 ledge of the cube root of this expression, let us inquire 

 by what steps it may be evolved. 



We see that the first term a of the root is at once 

 obtained from the leading term of the polynomial ; wo 

 thus have the step 



a 3 + 3a 2 6 + 3a6 + lr\a 



3a'6 + 



and regarding the remainder, hero exhibited, as a divi- 

 dend, it remains to find a divisor of it, such that the 

 quotient may be 6. It is plain that three times the 

 square of the root term a, just found, taken as a trial 

 or incomplete divisor, suffices to suggest the second term 

 6 of the root ; and we see, moreover, that if three times 

 the product of the two root terms, o and 6, as also the 

 square of 6, be added to the trial divisor 3o s , that the 



Tho following is an example of the operation : 



complete divisor, corresponding to tho quotient /, will 

 bo obtained ; hcnoo tho hiiishud process is as follows : 



1 3a'6 + 3a5 + 6 s 



If the root consist of three terms, a + ft-f-e, that is, if 

 the polynomial be (a + 6 + c) J , or (<< -j- //)' + 3 (a -J 

 + 3 (a + fcX* + * tnen tl>e portion (a -f I) 3 of this 

 polynomial may be exhausted, as above, :m.l tin- first 

 two terms o + b of the root thus found. And, just as 

 in the former case, 6 was derived from a, so here it is 

 plain that c may be derived from a -f 6. From these 

 considerations the following rule is suggested : 



RULE. Arrange the terms according to tho powers of 

 one of tho letters, as in the operation for the square 

 root. Put tho cube root of tho leading term in tho quo- 

 tient's place, subtract the cube of it from the polynomial, 

 bringing down the next three terms for a dividend, to 

 the left of which mark off a place for the divisor. In 

 this place put three times the square of tho root tnn just 

 found ; this will be the trial divisor, and the quotient it 

 suggests will be the second term of tho root. 



To three times the product of this new term and tho 

 preceding, add tho square of the new term, and connect 

 the .result to the trial divisor ; the whole will be the 

 complete divisor. 



Multiply the complete divisor by the new term, sub- 

 tract tho product from the dividend, and annex throe 

 more terms of the polynomial to the remainder ; the 

 whole will be the second dividend. 



For tho corresponding trial divisor put three times the 

 square of tho root-quantity now found, in the divisor's 

 place ; and with this trial divisor find tho third term of 

 the root ; and then complete the divisor, by adding to 

 what is already in the divisor's place three times the 

 product of the new term and the preceding part of the 

 root, and also the square of tho now term ; proceed tln'ii 

 as in division, adding three new terms of the polynomial 

 to the remainder ; and so on, till all tho terms of the 

 polynomial have been brought down. 



You will perceive that the principal part of the work 

 consists informing the successive divisors, each of which 

 ia made up of three portions namely, thrice the square 

 of the part of the root previously obtained, thrice tho 

 product of this part and the new term, and the square of 

 that new term. The first of these portions is what wo 

 have called the trial divisor; but you will not fail to 

 notice, that, as in the operation for the square root, tlio 

 leading term of tho first divisor is always sufficient to 

 make known any subsequent term of the root ; so that 

 after the first divisor is obtained, every new tenn of tho 

 root may be discovered at once, without tho aid of any 

 special trial divisor ; and therefore every subsequent 

 divisor may be inserted at once, in its proper place, u:ul 

 in its complete form. 



Rr + 15*' 20*' + 15*' Or + 1 (* 2x + I 



' 6*> + 4* > ) ' 



6*4-15* 



G* 5 -(- 12*' 8jr 



1ZH 4- 15.r 6* -f 

 This U made up of 



3*' I2x* + 15*' 6* 

 3.r _ 12* 3 -f 15x 



Aa a second example the following may bo taken 



27*' 54* + 63*' - 44** + 21*' - 6* + 1 

 27j* 



54* -}- 63*< 



2x -f 1 



27** 3C* 1 + 21*' 6* + 1) 27.r' 36* 4- 21*' C* + 1 

 Tliis U made up of i!7 - ' 



2*)' + 3 (3* 1 2*) -f 1 



