CUBE ROOT 



1664 



CUBE ROOT 



Cube Root of Numbers Consisting of id's and 

 Units. We shall see first how such numbers 

 are cubed and of what their cubes consist ; then 

 we can understand how to find the cube root 

 of a number. 



Let us cube 26. 

 26 = 20 + 6. 

 26=(20 + 6), or 

 26=(20 + 6)X(20 + 6)X<20 + 6). 



20 +6 

 20 +6 



20'+(20X6) 

 + (20x6)+6' 



20 J + 2X(20X6)+6 J 

 20 + 6 



20+2X (20*X6) + (20x6 J ) 

 (20 2 X6)+2x(20x6 2 )+6 



20+3X (20 2 X6)+3X (20x6 2 )+6* 

 Therefore : 



26=20>+(3X20 2 X6) + (3x20x 



Translating this into words, we have : "The 

 cube of a number is made up of the cube of the 

 tens, and 3 times the square of the tens times the 

 units, and 3 times the tens times the square of the 

 units, and the cube of the units." 



The figure below shows these parts together as 

 the whole cube. 



tV3(t 2 u)t3(tu 2 )tu 3 



THE COMPLETED CUBE 



Following this truth we write out the cube of 

 38, and have: 38 3 =30 8 + (3x30 2 x8) + (3X30X 

 8 2 )+8 3 . 



Putting the general truth into concise form, we 

 have: ( tens + units ) 3 = tens 8 + (3Xtens 2 x units) + 

 ( 3 X tens x units 2 ) + units 3 . 

 Or, using t for tens and u for units, we have : 



From this we are able to find the cube root of a 

 nuiriber, as below : 



Let us take 941192. Some number has been 

 cubed to give this. It is indicated thus ; 



n s = 941192 



The process of finding the cube root : 

 90 + 8 



941192 = t+(3Xt'Xw) +(3 XXw*)+tt 

 729000 = *" 



3X90 2 = 

 24300 



17792=(3XXM 2 )+W 



512 = 8 



"What is the largest cube of tens in the num- 

 ber?" Thinking of 941000 (which contains the 

 10's cubed) as 941, we find 729, or 9 8 , the largest 

 cube. Nine is the cube root of this, but 9 tens is 

 90, and 90 S = 72900, which we take out; the re- 

 mainder, 212192, must contain the rest of the 

 cube, namely (3xt 2 XM) + (3xtXM 2 )+ u*. Since 

 the first term, 3 X t 2 X u, is much the largest 

 part of the remainder, we may for the moment 

 consider 212192 = 3xt 2 X. We have the product 

 of three numbers and have two known numbers, 

 3 and 90 2 , to find u, the one not known, as ex- 

 pressed in the following : 



= 3X8100XM. 

 Therefore, to find u, we divide 212192 by 24300, 

 and find u = S. Now we take out 3Xt 2 Xu or 

 3X90 2 X8, and have left 17792, which must 

 contain (3XX 2 )+w 8 , the rest of the cube. We 

 take out 3X*XM 2 or 3X90x64 or 17290, and 

 have left 512, which equals 8 3 or u". So we 

 have found that 941192 is the cube of 98 or 

 ^941192 = 98. This is the simplest method for 

 those not familiar with algebra. For more ad- 

 vanced students the problem may be worked as 

 follows : 



90 + 8 



94TT92 = t 3 + 3 2 w + 3 tu* + M 3 

 729000 = t 8 



3t 2 =24300 



3tu= 2160 



M 2 = 64 



212192= 

 212192 



26524 



First find the largest cube of tens, which is 

 seen to be 729, and the cube root, 9 tens or 90 ; 

 take out 90 3 , and there is left 212192, which must 

 contain the other three terms of the cube ; that is, 



212192 = 3t*M + 3fw !! + M 8 , or, 

 212192= (3 2 + 3tM + M 2 )w. 



Since 3( 2 M is much the largest part of the three 

 quantities, we may for a moment neglect the 

 other two and have 212192 = 3t 2 w. We have the 

 product of three numbers and two of them given 

 to find the third. We find u by dividing 212192 

 by 3X90 2 and get u S. Then we substitute 90 

 for t and 8 for u in each part inside the paren- 

 theses and get the entire divisor as shown. 26524. 

 This multiplied by u, or 8, gives 212192. This 

 shows that we find in 941132 the cube of 98. 

 Therefore, 98 is the cube root of 941192, or 

 \X941192 = 98. 



The usual method followed after the subject is 

 understood is given below accompanied by an 

 illustration : 



