# Square and Cube Mathematics

Introduction:

During the Mathematical Calculation , candidates have to calculate the square and cubes of different numbers. Hence, the maximum practice of short-cut methods or tricks is very important for candidates. Therefore, it is advisable to use short-cut methods to calculate the square of numbers greater than 50 and cube of numbers greater than 20 numbers, but remembering square of numbers up to 50 and cube of numbers up to 20 Will be more lenient.

There are two methods for calculating squares of different numbers and cubes.

(i) General Rule – This method allows the square and cubes of any number to be calculated.

(ii) Special Rule – This method extends the square or cube of any number following a particular condition.

### (A) Calculate square (TO FIND OUT SQUARE)

Normal method

TYPE-1 To the Square of two-digit numbers.

Trick → if AB is the number of two digits

(AB)2 = A2/2AB/B2

Example: (64)2 =?

Solution: (64)2 = 62 / 2 × 6 × 4 / 42 = 36 48,6 = 4096 Ans.

Note: Here 1 and 4 (From the right) are the numbers obtained from carry of 16 and 48 respectively.

TYPE-2 To find the square of three-digit numbers

TRICK → If ‘A B C’ is a number of three digits, then it can be calculate as follows –

(ABC)2 = A2 /2.ab/2.ac + B2 /2.BC/ C2

Example: (346)2 =?

Solution: (346)2 = 32 / 2 × 3 × 4/2 × 3 × 6 + 42 / 2 × 4 × 6/62

Note: Here 3, 4, 5 and 2 (Carry) are the numbers obtained from 36, 48, 52 and 24 respectively.

Exercise Resolve by tricks –

1. (437)2 =?
2. (543)2 =?
3. (724)2 =?
4. (836)2 =?
5. (358)2 =?

TYPE-3 to find the square of four-digit numbers:

Trick → If ‘AB C D’ is a number of four digits, then it can be Calculated as follows

(ABCD)2 =?

1st step. = D2
2nd step. = 2.C.D
3rd step. = 2.B.D + C2
4th step. = 2.A.D + 2.B.C
5th step. = 2.A.C + B2
6th step. = 2.A.B
Last Step. = A2

Example: (6324)2 =?

Solution :
1st step. = 42 = 16 ⇒ 16
2nd step. = (2 × 2 × 4) = 16 ⇒ 16
3rd step. = (2 × 3 × 4) +22 = 28 ⇒ 28
4th step. = (2 × 6 × 4) + (2 × 3 × 2) = 60 ⇒ 60
5th step. = (2 × 6 × 2) + 32 = 33 ⇒ 33
6th step. = (2 × 6 × 3) = 36 ⇒ 36
last step. = 62 = 36

∴ ? = 363633281616 = 39992976 Ans.

Note: Here 1, 1, 2, 6, 3 and 3 (from the right) are the numbers obtained from carry of 16, 16, 28, 60, 33 and 36 respectively.

Exercise Solve them by TRICK-

1. (4567)2 =?
2. (8163)2 =?
3. (7435)2 =?
4. (3462)2 =?
5. (7246)2 =?

TYPE-4 To find the square of five-digit numbers

TRICK → If ‘A B C D E’ is a number of five digits, then it can be Calculated as follows

(ABCDE)2 =?
1st step. = E2
2nd step. = (2 × D × E)
3rd step. = (2 × C × E) + D2
4th step. = (2 × B × E) + (2 × C × D)
5th step. = (2 × A × E) + (2 × B × D) + C2
6th step. = (2 × A × D) + (2 × B × C)
7th step. = (2 × A × C) + B2
8th step. = (2 × A × B)
last step. = A2

Example (32457)2 =?
Solution:
1st step. = 72 = 49 ⇒ 49
2nd step. = (2 × 5 × 7) = 70 ⇒ 70
3rd step. = (2 × 4 × 7) + 52 = 81 ⇒ 81
4th step. = (2 × 2 × 7) + (2 × 4 × 5) = 68 ⇒ 68
5th step. = (2 × 3 × 7) + (2 × 2 × 5) + 42 = 78 ⇒ 78
6th step. = (2 × 3 × 5) + (2 × 2 × 4) = 46 ⇒ 46
7th step. = (2 × 3 × 4) = 22 = 28 ⇒ 28
8th step. = (2 × 3 × 2) = 12 ⇒ 12
Last step. = 32 = 9

∴ ? = 912284868817049 = 953456849 Answer.

Note: Here 4, 7, 8, 6, 7, 4, 2 and 1 (from the right) are the numbers obtained from carry of 49, 70, 81, 68, 78, 46, 28 and 12 respectively.

Special Rule

To calculate the square of any number made from a repeating number of the same digit (To find the square of a repeat digit number)

TYPE-1 to find the square of 1 – repeated number.

In order to calculate the square of numbers made from the repetition of the number 1, the number of times the number ‘1’ is present, write the number 1, 2, 3. in the increasing order from the right to the left. and 2,1 in decreasing order to calculate the square of given number.

Example 1. (111)2 = 12321 = 12321. Ans.
Example 2. (1111)2 = 1234321 = 1234321 Ans.
Example 3. (11111)2 = 123454321 = 123454321 Ans.

Exercise Solve them by TRICK-

1. (11)2 =?
2. (111111)2 =?
3. (1111111)2 =?
4. (1111.111)2 =?
5. (111111.11)2 =?

TYPE – 2 Find out the square of 2, 3, 4, 5, 6, 7, 8 or 9-repeated numbers.

If the number of numbers made from iterations of 2, 3, 4, 5, 6, 7, 8, or 9,

TRICK → the number of times the number of digits would be present in the Given number. Write the digits 3,2,1 in the ascending order from right to left and 1,2 in the descending order, then multiply it with the square of repeated digit and the number obtained after multiplication is the square of the Given number. So the square of repeat digit

Example 1. (222)2 = 22 × (12321) = 4 × (12321) = 49284 Answer
Example 2. (4444)2 = 42 × (1234321) = 16 × (1234321) = 19749136 Answer.
Example 3. (777)2 = 72 × (12321) = 49 × (12321) = 603729 Answer.

Special tricks in case of 3, 6 or 9-repeat digit numbers

TRICK → With the help of ‘O Z E N’ , the square of the numbers formed from the repetition of the number 3 is computed. Here Z (0) and N (9) are always kept constant i.e. written only once and the number of O (1) and E (8) is written one less than times the repeated digit present in the given number.

Example 1. (3333)2 = 11108889 Answer.
Note: Here there are four digits in 3333. Therefore, 1 and 8 have been written three or three times.

Example 2. (333.33)2 = 111108.8889 Answer.

TRICK → With the help of  ‘FO T FI S’ , the square of the numbers made from the repetition of number 6 is drawn. Here T (3) and S (6) always remain Constant. That is, written only once and the number of times FO (4) and FI (5) is written one less than times the repeated digit ‘6’ is present in the given number.

Νοte: Here’s the four digits in 6666. Therefore 4 and 5 are written three times.

TRICK → With the help of ‘NEZO’ , the square of the numbers made from the repeatition of number 9 is drawn. Here E (8) and O (1) always remain Constant. That is, written only once and the number of times And the number of times N (9) and Z (0) is written one less than times the repeated digit ‘9’ is present in the given number.

Example 1. (999)2 = 998001. Answer.

Note: Here are three digits in 999 Therefore 9 and 0 have been written twice

Example 2. (999.99)2 = 999980.0001 Answer

TYPE – 3 To find out the square of numbers in the neighbourhood of 10, 100, 1000 and 10000

TRICK → To compute the square of numbers nearest to 10, 100, 1,000 and 10,000 numbers, if the number to be calculated from the number of squares is less than one of these numbers, then the square is written on the left hand side of the number after subtracting it these numbers. And the number to be subtracted is written on the right side of the square.

Example 1 . (8)2 = 8 – 2/22 = 64 Answer.

Example 2 . (114)2 = 114 + 14 / (14)2 = 128196 = 12996 Answer.

Note: If the square of number nearest to 100 is to be computed, then compute the square of the number to be subtracted and its two digits are kept on the right side.

Example 3. (978)2 = 978 – 22 / (22)2 = 956/484 = 956484 Answer.

Example 4. (10032)2 = 10032 + 32/322 = 10064/1024 = 100641024 Answer.

Exercise Solve them by TRICK-

1. (87)2 =?
2. (124)2 =?
3. (972)2 =?
4. (1026)2 =?
5. (9982)2 =?
6. (9974)2 =?
7. (10023)2 =?
8. (10035)2 =?

Remember the class of these numbers to resolve quickly

### (B) To find the cube

Simple Method

The two-digit numbers of the two digit numbers

TRICK → Cube of a number of two digits is determined by the following methods:
(i) First, calculate the cube of the tens place of the two digits number and placed it on the left side at the first position.
(ii) The ratios are made in tens and unit digits.
(iii) By dividing the first term with the ratio of tenth, multiplying the quotient divided by the ratio of the unit, the number received is placed on the right side of the first term as the second term. The third and fourth digits are also written by this type of action.
(iv) Second and third digits are doubled and written down in them.
(v) One digit is dropped from the right side and the remaining number is added to its left as carry.

Example 1. (35)3 =?

Solution:
1st step. = 33 = 27
2nd step. = 3 : 5
3rd step. = 27 × 5 / 3 = 45
4th step. = 45 × 5 / 3 = 75
5th step. = 75 × 5 / 3 = 125
last step. = (45 × 2) = 90 and (75 × 2) = 150

Example 2. (93)3 =?

Solution:
1st step. = 93 = 729
2nd step. = 9 : 3 = 3 : 1
3rd step. = 729 × 1 / 3 = 243
4th step. = 243 × 1 / 3 = 81
5th step. = 81 × 1 / 3 = 27
last step. = (243 × 2) = 486 and (81 × 2) = 162

Exercise – Solve them by TRICK

1. (16)3 =?
2. (24)3 =?
3. (48)3 =?
4. (56)3 =?
5. (83)3 =?

Special Rules

To find out the cube of 1, 2, …… 9 repeated numbers.

TRICK → Cube of a repeating number is obtained by multiplying that number with the square of that number.

Example 1 . (111)3 = (111)2 × 111 = 12321 × 111

= 1 / (1 + 2) / (1 + 2 + 3) / (2 + 3 + 2) / (3 + 2 + 1) / (2 + 1) / 1 = 1367631 Answer

Example 2. (222)3 = (222)2 × 222 = 49284 × 222

= 4 × 2 / (4 + 9) × 2 / (4 + 9 + 2) × 2 / (9 + 2 + 8) × 2 / (2 + 8 + 4) × 2 / (8 + 4) × 2 / 4 × 2

Exercise – Solve them by TRICK

1. (22)3 =?
2. (333)3 =?
3. (444)3 =?
4. (55)3 =?
5. (66)3 =?
6. (77)3 =?
7. (88)3 =?
8. (99)3 =?
9. (1111)3 =?

Remember the cube of these numbers for a quicker resolution