Cubes by Bhārati Kṛṣṇa Tīrtha

Introduction

Here we will be learning how to find the cubes of two digit numbers. We will be using the sūtra

आनुरूप्येण

" Ānurūpyeṇa "

" Proportionally "

The detailed steps are explained below .

Let a and b are the two digits .

Step 1 : We will be writing the four numbers in geometric ratio in exact proportion.

First term will be $a^{3}$ , common ratio = ${\frac {b}{a}}$

second term will be $a^{3}\ X\ {\frac {b}{a}}=a^{2}\ X\ b$

third term will be $a^{2}b\ X\ {\frac {b}{a}}=a\ X\ b^{2}$

fourth term will be $ab^{2}\ X\ {\frac {b}{a}}=b^{3}$

	First term	Middle terms (second, third terms)		Fourth term
Step 1	$a^{3}$	$a^{2}\ X\ b$	$a\ X\ b^{2}$	$b^{3}$
Step 2	$a^{3}$	double of the middle terms $2(a^{2}\ X\ b)$	double of the middle terms $2(a\ X\ b^{2})$	$b^{3}$
Step 3	$a^{3}$	add the two middle terms $a^{2}\ X\ b+2(a^{2}\ X\ b)$	add the two middle terms $a\ X\ b^{2}+2(a\ X\ b^{2})$	$b^{3}$
Final Step	$a^{3}$	$3(a^{2}\ X\ b)$	$3(a\ X\ b^{2})$	$b^{3}$

This is nothing but the expansion of the formula for $(a+b)^{3}$ where a and b are two individual digits of the number. This will makes the calculation of cubes very fast and easy. This will be explained through the below examples.

Cubes of natural numbers

1	2	3	4	5	6	7	8	9
1	8	27	64	125	216	343	512	729

Example: 23³

Here a = 2 , b = 3 , Using the final step mentioned above.

$a^{3}$	$3(a^{2}\ X\ b)$	$3(a\ X\ b^{2})$	$b^{3}$
23 = 8	3 X ( 22 X 3) = 3 (4 X 3)	3 X (2 X 32) = 3 X (2 X 9)	33 = 27
8	36	54	27
8	36	54	Put 7 and carry over 2
8	36	54 + Carry over (2)	7
8	36	56	7
8	36	Put 6 and carry over 5	7
8	36 + Carry over (5)	6	7
8	41	6	7
8	Put 1 and carry over 4	6	7
8 + Carry over (4)	1	6	7
12	1	6	7

Answer : 23³ = 12167

Example: 12³

Here a = 1 , b = 2 , Using the final step mentioned above.

$a^{3}$	$3(a^{2}\ X\ b)$	$3(a\ X\ b^{2})$	$b^{3}$
13 = 1	3 X ( 12 X 2) = 3 (1 X 2)	3 X (1 X 22) = 3 X (1 X 4)	23 = 8
1	6	12	8
1	6	Put 2 and carry over 1	8
1	6 + Carry over (1)	2	8
1	7	2	8

Answer : 12³ = 1728

Example: 25³

Here a = 2, b = 5 , Using the final step mentioned above.

$a^{3}$	$3(a^{2}\ X\ b)$	$3(a\ X\ b^{2})$	$b^{3}$
23 = 8	3 X ( 22 X 5) = 3 (4 X 5)	3 X (2 X 52) = 3 X (2 X 25)	53 = 125
8	60	150	125
8	60	150	Put 5 and carry over 12
8	60	150 + Carry over (12)	5
8	60	162	5
8	60	Put 2 and carry over 16	5
8	60 + Carry over (16)	2	5
8	76	2	5
8	Put 6 and carry over 7	2	5
8 + Carry over (7)	6	2	5
15	6	2	5

Answer : 25³ = 15625