** Introduction: **

During the Mathematical Calculation , competitors are often required to remove classes and cubes of different numbers. Hence, the maximum practice of short-cut methods or tricks for competitors is very important. Therefore, it is advisable to use short-cut methods to remove the cube of greater than 50 numbers and larger 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 removing squares of different numbers and cubes.

** (i) General Rate –** This method allows the class and cubes of any number to be removed.**
(ii) Special Rule –** This method extends the square or cube of any number following a particular condition.

** (A) Remove 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} = A^{2}/2AB/B^{2}

** Example: **(64)^{2} =?

** Solution: **(64)^{2 }= 6^{2} / 2 × 6 × 4 / 4^{2} = 36 48,6 = 4096 **Ans.**

** Note: **Here 1 and 4 (From the right) are the numbers of the numbers obtained from 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 sorted out as follows –

(ABC)^{2} = A^{2} /2.ab/2.ac + B^{2} /2.BC/ C^{2}

**Example: **(346)^{2} =?

** Solution: **(346)^{2} = 3^{2} / 2 × 3 × 4/2 × 3 × 6 + 4^{2} / 2 × 4 × 6/6^{2 }

** Note:** are 3, 4, 5 and 2 (Carry) numbers 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 sorted out as follows

(ABCD)^{2} =?

** 1st step. **= D^{2}

** 2nd step. **= 2.C.D

** 3rd step. = **2.B.D + C^{2}

** 4th step. = **2.A.D + 2.B.C

** 5th step. = **2.A.C + B^{2}

** 6th step. = **2.A.B

** Last Step. = **A^{2}

** Example: **(6324)^{2} =?

** Solution :
1st step. **= 4

∴ ? = 36_{3}6_{3}3_{2}8_{1}6_{1}6 = 39992976 **Ans.**

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

** 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 sorted out as follows

(ABCDE)^{2} =?

**1st step.** = E^{2}

**2nd step.** = (2 × D × E)

**3rd step.** = (2 × C × E) + D^{2}

**4th step.** = (2 × B × E) + (2 × C × D)

**5th step.** = (2 × A × E) + (2 × B × D) + C^{2}

**6th step.** = (2 × A × D) + (2 × B × C)

**7th step.** = (2 × A × C) + B^{2}

**8th step.** = (2 × A × B)

**last step.** = A^{2 }

**Example **(32457)^{2} =?

**Solution:**

** 1st step. **= 7

∴ ? = 9_{1}2_{2}8_{4}8_{6}8_{8}1_{7}0_{4}9 = 953456849 ** Answer. **

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

** Special Rule **

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

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

In order to remove the square of numbers made from the repetition of the number 1, the number of times the number ‘1’ is in the number 1, 2, 3. in the increasing order from the right to the right. In decreasing order. The class of numbers given by typing 2, 1 is obtained.

** Example 1. **(111)^{2} = 12321 = 12321. **Ans.**

** Example 2. **(1111)^{2} = 1234321 = 1234321 **Ans.
Example 3. **(11111)

** 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 answer number As much as 3,2,1 in the moving order and 1,2 in the descending order, on the right side, multiply it into the square of repeat digit and the number obtained on the right is multiplied. Adds it to the number on the left side. The product so obtained is the square of the answer number. So the square of repeat digit

** Example 1.** (222)^{2} = 2^{2} × (12321) = 4 × (12321) = 49284 **Answer**

** Example 2. **(4444)^{2} = 4^{2} × (1234321) = 16 × (1234321) = 19749136** Answer.
Example 3. **(777)

** 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 iterative of the number 3 is drawn. Here Z (0) and N (9) are always kept constant i.e. once written and the number of O (1) and E (8) is presently known in the repeated scores, there is less one right than that

** 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 recurrence of a score 6 is drawn. Here T (3) and S (6) always remain Constant. That is, once written and the number of times FO (4) and F1 (5) is in the number of times, in the number of repeat digit ‘6’, there is one less than that.

** Example: **(6666)^{2} = 44435556 **Answer**

** Νοte: **Here’s the four digits in 6666. Therefore 4 and 6 are written three times.

**TRICK → **With the help of ” NEZO , the square of the number made from the numerator 9 is always constant, i.e. written once. And the number of times N (9) and Z (0), repeat digit number 9, the number of times the presence occurs, is less than the right one.

** Example 1. **(999)^{2} = 998001. **Answer.**

** Note: **Here is the 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 extract a square of 10, 100, 1,000 and 10,000 numbers, if the number to be extracted from the number of squares is less than one of these numbers, then the squared is written down on the left hand side of the number to be extracted is. And the number to be subtracted is written by the class on the right side.

** Example 1 .** (8)^{2} = 8 – 2/2^{2} = 64 **Answer.**

** Example 2 .** (114)^{2} = 114 + 14 / (14)^{2} = 128_{1}96 = 12996 **Answer.**

** Note: **If the number of close numbers of 100 is to be removed, then the number to be subtracted and its two digits are kept on the right side.**
Example 3. **(978)

** Example 4. **(10032)^{2} = 10032 + 32/32^{2} = 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

**Example 1.** (35)^{3} =?

** Solution:
1st step. **= 3

**Example 2. **(93)^{3} =?

** Solution:
1st step. **= 9

**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 recurring number is obtained by multiplying that number into that square by 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

= 8_{2}6_{3}0_{3}8_{2}848 = 10941048 **Answer.**

** 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 **

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