Cube in Sadratnamālā

Here we will be knowing how to find cube as per Sadratnamālā.

Verse

ज्या हृत् सीरी भाति शरण्यः तत्पुरि गूढाङ्गः श्रीकृष्णः ।

धीरोऽसौ एकादिनवान्तं तुल्यत्र्यभ्यासः अत्र घनः स्यात् ॥ १५ ॥

As per Kaṭapayādi Notation assign the numbers to the Sanskrit alphabets of the above verse.

ज् या	हृ त्	सी री	भा ति	श र ण्यः	तत् पु रि	गू ढा ङ्गः	श्री कृ ष्णः	धी रो सौ
ज् य् आ	ह् ॠ त्	स् ई र् ई	भ् आ त् इ	श् अ र् अ ण् यः	त त् प् उ र् इ	ग् ऊ ढ आ ङ् गः	श् र् ई क् ॠ ष् णः	ध् ई र् ओ स् औ
- 1 -	8 - -	7 - 2 -	4 - 6 -	5 - 2 - - 1	6 - 1 - 2 -	3 - 4 - - 3	- 2 - 1 - - 5	9 - 2 - 7 -
1	8	72	46	521	612	343	215	927
1	8	27	64	125	216	343	512	729	अङ्कानाम् ‌वामतो गतिः numbers go from right to left

Translation

One, eight, twenty-seven, sixty-four, one hundred and twenty-five, two hundred and sixteen, three hundred and forty-three, five hundred and twelve, seven hundred and twenty-nine are the cubes of the numbers from one to nine. The product of three (equal numbers) is the cube (of that number).

Verse

घनेऽथ तन्मूलवर्गतदादि त्रिवधे ततः ।

आदिवर्गान्त्यत्रिवधे युतेष्वङ्केष्वथो घनः ॥ १६ ॥

Translation

To the cube (of the last digit) add (on the right) the product of thrice the square of the last digit and the remaining digits. Then add (on the next place to the right) the product of thrice the last digit and the square of the remaining part and then add the cube of the remaining part (on the next place to the right). This is (repeated until all the digits are finished to get) the cube.

Example: Cube of 35

Here the number considered is 35 , 3 is the last digit and the remaining digit is 5. Apply the method of cubing.

Cube of the last digit (33) = 27	2	7
Thrice the square of the last digit (3) multiplied by the remaining digit (5) 3 X 32 X 5 = 135. To be placed in the next place such that 5 is one place next to 7	1	3	5
Thrice the square of the remaining digit (5) multiplied by the last digit (3) 3 X 52 X 3 = 225. To be placed in the next place such that 5 is one place next to 5		2	2	5
Cube of the remaining digit (53) = 125. To be placed in the next place such that 5 is one place next to 5			1	2	5
Cube of 35 = 42875 ( adding each column value)	4	2	8	7	5

Cube of 35 = 42875

Example: Cube of 123

Here the number considered is 12 , 1 is the last digit and the remaining digit is 2. Apply the method of cubing.

Cube of the last digit (13) = 1	1
Thrice the square of the last digit(1) multiplied by the remaining digit (2) 3 X 12 X 2 = 6. To be placed in the next place		6
Thrice the square of the remaining digit (2) multiplied by the last digit (1) 3 X 22 X 1 = 12. To be placed in the next place such that 2 is one place next to 6		1	2
Cube of the remaining digit (23) = 8. To be placed in the next place				8
Cube of 12 = 1728 ( adding each column value)	1	7	2	8

Now take the remaining digits of 123 (digit after 12, that is 3) and apply the method of cubing. The number considered is 123 .Here the last digit is 12 and remaining digit is 3.

Cube of the last digit (123) = 1728 (from the above calculation)	1	7	2	8
Thrice the square of the last digit(12) multiplied by the remaining digit (3) 3 X 122 X 3 = 1296. To be placed in the next place such that 6 is one place next to 8		1	2	9	6
Thrice the square of the remaining digit (3) multiplied by the last digit (12) 3 X 32 X 12 = 324. To be placed in the next place such that 4 is one place next to 6				3	2	4
Cube of the remaining digit (33) = 27. To be placed in the next place such that 7 is one place next to 4						2	7
Cube of 123 = 1860867 ( adding each column value)	1	8	6	0	8	6	7

Cube of 123 = 1860867

Example: Cube of 234

Here the number considered is 23, 2 is the last digit and the remaining digit is 3. Apply the method of cubing.

Cube of the last digit (23) = 8	8
Thrice the square of the last digit (2) multiplied by the remaining digit (3) 3 X 22 X 3 = 36. To be placed in the next place such that 6 is one place next to 8	3	6
Thrice the square of the remaining digit (3) multiplied by the last digit (2) 3 X 32 X 2 = 54. To be placed in the next place such that 4 is one place next to 6		5	4
Cube of the remaining digit (33) = 27. To be placed in the next place such that 7 is one place next to 4			2	7
Cube of 23 = 12167 (adding each column value)	12	1	6	7

Now take the remaining digits of 234 (digit after 23, that is 4) and apply the method of cubing. The number considered is 234. Here the last digit is 23 and remaining digit is 4.

Cube of the last digit (233) = 12167 (from the above calculation)	1	2	1	6	7
Thrice the square of the last digit(23) multiplied by the remaining digit (4) 3 X 232 X 4 = 6348. To be placed in the next place such that 8 is one place next to 7			6	3	4	8
Thrice the square of the remaining digit (4) multiplied by the last digit (23) 3 X 42 X 23 = 1104. To be placed in the next place such that 4 is one place next to 8				1	1	0	4
Cube of the remaining digit (43) = 64. To be placed in the next place such that 4 is one place next to 4							6	4
Cube of 234 = 12812904 (adding each column value)	1	2	8	1	2	9	0	4

Cube of 234 = 12812904

Example: Cube of 5678

Here the number considered is 56, 5 is the last digit and the remaining digit is 6. Apply the method of cubing.

Cube of the last digit (53) =125	1	2	5
Thrice the square of the last digit (5) multiplied by the remaining digit (6) 3 X 52 X 6 = 450. To be placed in the next place such that 0 is one place next to 5		4	5	0
Thrice the square of the remaining digit (6) multiplied by the last digit (5) 3 X 62 X 5 = 540. To be placed in the next place such that 0 is one place next to 0			5	4	0
Cube of the remaining digit (63) = 216. To be placed in the next place such that 6 is one place next to 0				2	1	6
Cube of 56 = 175616 (adding each column value)	1	7	5	6	1	6

Now take the remaining digits of 5678 (digit after 56 , that is 7) and apply the method of cubing. The number considered is 567. Here the last digit is 56 and remaining digit is 7.

Cube of the last digit (563) = 175616 (from the above calculation)	1	7	5	6	1	6
Thrice the square of the last digit (56) multiplied by the remaining digit (7) 3 X 562 X 7 = 65856. To be placed in the next place such that 6 is one place next to 6			6	5	8	5	6
Thrice the square of the remaining digit (7) multiplied by the last digit (56) 3 X 72 X 56 = 8232. To be placed in the next place such that 2 is one place next to 6					8	2	3	2
Cube of the remaining digit (73) = 343. To be placed in the next place such that 3 is one place next to 2							3	4	3
Cube of 567 = 182284263 (adding each column value)	1	8	2	2	8	4	2	6	3

Now take the remaining digits of 5678 (digit after 567 , that is 8) and apply the method of cubing. The number considered is 5678. Here the last digit is 567 and remaining digit is 8

Cube of the last digit (5673) = 182284263 (from the above calculation)	1	8	2	2	8	4	2	6	3
Thrice the square of the last digit (567) multiplied by the remaining digit (8) 3 X 5672 X 8 = 7715736. To be placed in the next place such that 6 is one place next to 3				7	7	1	5	7	3	6
Thrice the square of the remaining digit (8) multiplied by the last digit(567) 3 X 82 X 567 = 108864. To be placed in the next place such that 4 is one place next to 6						1	0	8	8	6	4
Cube of the remaining digit (83) = 512 To be placed in the next place such that 2 is one place next to 4										5	1	2
Cube of 5678 = 183056925752 (adding each column value)	1	8	3	0	5	6	9	2	5	7	5	2

Cube of 5678 = 183056925752

See Also

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