Cube root in Līlāvatī

Introduction

Here we will know how to find the cube root of a number as mentioned in Līlāvatī.

Verse no. 28 :

आद्यं घनस्थानमथाघने द्वे

पुनस्तथान्त्याद् घनतो विशोध्य ।

घनं पृथक्स्थं पदमस्य कृत्या

त्रिघ्न्या तदाद्यं विभजेत् फलं तु ॥ 28 ॥

Verse no. 29 :

पङ्क्त्या न्यसेत् तत्कृतिमन्त्यनिघ्नीं

त्रिघ्नीं त्यजेत्तत् प्रथमात् फलस्य ।

घनं तदाद्याद् घनमूलमेवं

पङ्क्तिर्भवेदेवमतः पुनश्च ॥ 29 ॥

Translation

Draw a vertical bar above the digit in the unit's place of the number whose cube root is required. Then put horizontal bars on the two digits to its left, vertical bar on the next and repeat until the extreme left hand digit is reached.

From the extreme left-hand section, deduct the highest cube possible and write to the left side the number (a) whose cube was subtracted. Write to the right of the remainder, the first digit of the next section to get new sub-dividend. Now divide by 3a² and write the quotient b next to a. Then write the next digit from the section to the right of the remainder obtained above. The next divisor is 3ab². In the next step take b³ as the divisor. Continue this procedure till the digits in the given number are exhausted.

Example: Cube root of 19683

		-	|	-	-	|	Steps: First we put the bars and horizontal lines. Start from unit's place with "|" put "-" for the 2nd and 3rd digit, "|" for the 4th digit."-" for the 5th digit .Grouping will be done till the "|". Hence the first group is 19 and the other group is 683. Deduct the highest cube (23 = 8) from 19.Remainder is 11 , write 2 in the root column and write the next digit which is 6. The number we got is 116. The new divisor is 3 x 22 = 12. We can go up to 12 X 9 =108 to subtract from 116. If we do, further subtractions will not be possible.Hence we take 7 as the quotient. 12 X 7 = 84. 116 - 84 = 32. Now we take the next digit which is 8. The number we got is 328.The new sub-divisor is 3 X 2 X 72 = 294 which is subtracted from 328 to get 34. We write the next digit which is 3. The number we got is 343. this is subtracted by 73 = 343 to get zero.Write 7 in the root column. Hence the cube root of 19683 = 27 (taking the roots in the order we got)
Root	Paṅkti	1	9	6	8	3
2	23 = 8		8
		1	1	6
	3 x 22 x 7		8	4
			3	2	8
	3 x 2 X 72		2	9	4
				3	4	3
7	73 = 8			3	4	3
				0	0	0

Answer: Cube root of 19683 = 27

Example: Cube root of 1953125

		|	-	-	|	-	-	|	Steps: First we put the bars and horizontal lines. Start from unit's place with "|" put "-" for the 2nd and 3rd digit, "|" for the 4th digit."-" for the 5th and 6th digit , "|" for the 7th digit. Grouping will be done till the "|". Hence the first group is 1 and the second group is 953 and the last group is 125 Deduct the highest cube (13 = 1) from 1.Remainder is 0 , write 1 in the root column and write the next digit which is 9. The number we got is 9. The new divisor is 3 x 12 = 3. We can go up to 3 X 3 =9 to subtract from 9. If we do, further subtractions will not be possible.Hence we take 2 as the quotient. 3 X 2 =6. 9 - 6 = 3. Now we take the next digit which is 5. The number we got is 35.The new sub-divisor is 3 X 1 X 22 = 12 which is subtracted from 35 to get 23. We write the next digit which is 3. The number we got is 233. this is subtracted by 23 = 8 to get 225.Take the next digit which is 1. The number we got is 2251. The new divisor is 3 X 122 . We got 12 by writing the root in the order we have got till now. (i.e 12). Here we take the quotient as 5 . 3 X 122 X 5 = 2160 which will be subtracted from 2251. The remainder is 91. Take the next digit 2 . The number we got is 912.The new sub-divisor is 3 X 12 X 52 = 900. which is subtracted from 912 to get 12. Take the next digit 5. The number we got is 125. this is subtracted by 53 = 125 to get zero.Write 5 in the root column. Hence the cube root of 1953125 = 125 (taking the roots in the order we got)
Root	Paṅkti	1	9	5	3	1	2	5
1	13 = 1	1
		0	9
	3 X 12 X 2		6
			3	5
	3 X 1 X 22		1	2
			2	3	3
2	23 = 8				8
			2	2	5	1
	3 X 122 X 5		2	1	6	0
					9	1	2
	3 X 12 X 52				9	0	0
						1	2	5
5	53 = 125					1	2	5
						0	0	0

Answer: Cube root of 1953125 = 125

See Also

लीलावती में 'घनमूल'