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Squares and Cubes (Vedic Mathematics)

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1. Finding Squares of Numbers Using General Method

This method of squaring is based on a corollary of “Nikhilam sutra”, which reads “Yavdunam Tavdunikratya vargam cha yojayet” and means “whatever the extent of its deficiency, lessen it further to that very extent; and also set up the square of its deficiency”.
This method is very helpful in calculating squares of numbers close to 10, 100, 1000, etc.
Three probable cases are possible:
Case I: When the number is less than and close to 10n (For example, 982)
Case II: When the number is greater than and close to 10n (For example, 10092)
Case III: When the number is not close to 10n (For example, 452)
Watch the following video to understand case I, where the number is less than 10n.
Solved Examples
Example 1:
Calculate 972
Solution:
Base = 100
Deficit = 100 − 97 = 3
32 = 09 … (1)
97 − 3 = 94 … (2)
∴972 = 9409
Example 2:
Calculate 9832
Solution:
Base = 1000
Deficit = 1000 − 983 = 17
172 = 289 … (1)
983 − 17 = 966 … (2)
∴9832 = 966289
The above examples involved calculating squares of numbers, which were less than and close to 10n. Now, consider the case, where the number is greater than 10n and close to it, i.e., numbers like 108, 1012, etc.
For this, watch the following video to understand the method.
Solved Examples
Example 1:
Calculate 1092
Solution:
Base = 100
Surplus = 109 − 100 = 9
92 = 81
109 + 9 = 118
∴1092 = 11881
Example 2:
Calculate 10212
Solution:
Base = 1000
Surplus = 1021 − 1000 = 21
212 = 441
1021 + 21 = 1042
∴10212 = 1042441
Note:
The only difference in the above two cases is that in the second case, the surplus was added to the number and was not subtracted. This can be summarized as:
When number < 10n: If x is the base and a is the deficit, then:
(Given number)2 = (x − a)2 = [(x − a) − a] a2
When number > 10n: If x is the base and a is the surplus, then:
(Given number)2 = (x + a)2 = [(x + a) + a] a2
However, what will happen if the square of the deficit is greater than the base, i.e., greater than the number 10n? Watch the following video to learn the method.
Solved Examples
Example 1:
Calculate 882
Solution:
Base = 100
Deficit = 100 − 88 = 12
122 = 144 > 100
⇒ Carry = 1
Part-II of 882 = 44
Part-I of 882 = 88 − 12 + Carry = 76 + 1 = 77
∴882 = 7744
Example 2:
Calculate 1212
Solution:
Base = 100
Deficit = 121 − 100 = 21
(21)2 = 441 > 100
⇒ Carry = 4
Part-II of 1212 = 41
Part-I of 1212 = 121 + 21 + Carry = 142 + 4 = 146
∴1212 = 14641
Notice that in all the above cases, the numbers were very close to the base 10n. But what if the number is not close to the base?
For example, consider the number 43 and try to find its square.
In this case, 43 lies between 10 and 100 and is closer to 10.
Thus, its base is taken as 10.
⇒ Surplus = 43 − 10 = 33
However, now, one needs to calculate the value of 332, which is not so simple.
A new concept is introduced in such cases, which is derived from the sub-sutra of Vedic Mathematics, called “Anurupayena”, which means “Proportionality”.
Watch the following video to understand this method.
Solved Examples
Example 1:
Calculate 422
Solution:
Base = 50 =
Deficit = 50 − 42 = 8
82 = 64
42 − 8 = 34
∴422 = = 1764
Example 2:
Calculate 5122
Solution:
Base = 500 =
Surplus = 512 − 500 = 12
122 = 144
512 + 12 = 524

2. Finding Squares of Numbers Ending With 5

There is a direct method for calculating the square of a number that ends with 5. Watch the following video to learn the method.
Solved Examples
Example 1:
Calculate 952
Solution:
{(First digit) × (First digit + 1)} = 9 × 10 = 90
∴952 = 9025
Example 2:
Calculate 652
Solution:
{(First digit) × (First digit + 1)} = 6 × 7 = 42
∴652 = 4225
Example 3:
Calculate 1152
Solution:
{(First two digits) × (First two digits + 1)} = 11 × 12 = 132
∴1152 = 13225
This method is a corollary of the “Nikhilam Sutra” and the “Ekadhiken Purven Sutra” of Vedic Mathematics.

3. Cubes of Two Digit Numbers

The cube of a number is calculated using the “Anurupaya Sutra” of Vedic Mathematics. However, for this, one needs to learn the cubes of the first nine natural numbers.
The cubes of single-digit natural numbers are:
13 = 1
23 = 8
33 = 27
43 = 64
53 = 125
63 = 216
73 = 343
83 = 512
93 = 729
Now, by using the above sutra, calculating the cubes of numbers from 11 to 99 becomes very easy. To understand the method, watch the
following video.
Solved Examples
Example 1:
Calculate 323
Solution:
Ratio =
323 = =
∴323 = 32768
Example 2:
Calculate 253
Solution:
It is easier to calculate the cube of 2 than to calculate the cube of 5. Hence, one should start from right to left taking the first term as 23 and the ratio of the last digit to the first digit.
Ratio =
(25)3 = =
∴253 = 15625

4. Cubes of Numbers Using General Method

Finding the cube of a number is a very difficult and time consuming process. With so many steps involved in multiplying a number, the possibility of making errors increases manifold.
This lesson involves finding the cube of a number, without actually multiplying the number thrice. This method is derived from the fifth sutra of Vedic Mathematics, which is known as the “Yavadunam Sutra”.
Here, two probable cases are possible:
Case I: When the number is close to and greater than 10n, for example, 102
Case II: When the number is close to and less than 10n, for example, 992
Watch the following video to understand case I:
Solved Example
Example 1:
Calculate 1133
Solution:
Base = 100
Surplus = 113 − 100 = 13
Part-III:
133 = 2197
Part-II:
13 × 2 = 26
13 + 26 = 39
39 × 13 = 507
Part-I:
13 × 2 = 26
113 + 26 = 139

∴1133 = 1442897
Now, case II deals with finding cubes of numbers that are less than and close to 10n. Its method only differs slightly from that of case I.
Hence, to understand case II, watch the following video.
Solved Example
Example 1:
Calculate 933
Solution:
Base = 100
Deficit = −7
Part-III:
(−7)3 = −343
Part-II:
−7 × 2 = −14
−7 + (−14) = −21
−21 × (−7) = 147
Part-I:
−7 × 2 = −14
93 + (−14) = 79

∴933 = 804357
Note:
This method is very useful to calculate the cubes of numbers that are very close to the base.

5. Fourth Power of a Two-Digit Number

The fourth power of any two-digit number can be calculated by using the “Anurupya sutra” of Vedic Mathematics.
Watch the following video to understand the method.
Solved Examples:
Example 1:
Calculate 154
Solution:
Ratio = 5: 1 = 5
Digit at tens place = 1
14 = 1
154 = =
∴154 = 50625
Example 2:
Calculate 444
Solution:
Ratio = 4: 4 = 1
Digit at tens place = 4
44 = 256
444 = =
∴444 = 3748096
Note:
This method is only valid to calculate the fourth power of a two-digit number.
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