In this blog, we will be going over how to find the Squares and Square roots of numbers

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Prerequisites

⭐️ You must by-heart the squares till 25.

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Square of Numbers:

Square of Numbers with base 50:

Numbers less than 50:

1. Get the difference between the given number and 50 and square it.

2. Subtract the same difference from 25.

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Example #1:

$$44^2 = ?$$

Step 1. Get the difference between the given number and 50 and square it.

$$50 - 44 = 6; 6^2 = 36$$

Step 2. Subtract the same difference from 25.

$$25 - 6 = 19;$$

Answer:

$$44^2 = 1936$$

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Example #2:

$$43^2 = ?$$

Step 1. Get the difference between the given number and 50 and square it.

$$50 - 43 = 7; 7^2 = 49$$

Step 2. Subtract the same difference from 25.

$$25 - 7 = 18;$$

Answer:

$$43^2 = 1849$$

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Example #3:

$$38^2 = ?$$

Step 1. Get the difference between the given number and 50 and square it.

$$50 - 38 = 12; 12^2 = 144$$

Step 2. Subtract the same difference from 25.

$$25 - 12 = 13;$$

⭐️ As Step 1 returned a 3-digit number and we can only occupy 2 digits in the end, we will take the third as carry and add it to the answer of Step 2

$$13 + 1 = 14$$

Answer:

$$38^2 = 1444$$

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Example #4

$$37^2 = ?$$

Step 1: Get the difference between the given number and 50 and square it.

$$50 - 37 = 13; 13^2 = 169$$

Step 2: Subtract the same difference from 25.

$$25 - 13 = 12$$

⭐️ As Step 1 returned a 3-digit number and we can only occupy 2 digits in the end, we will take the third as carry and add it to the answer of Step 2

$$12 + 1 = 13$$

Answer:

$$37^2 = 1369$$

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Numbers greater than 50:

1. Get the difference between the given number and 50 and square it.

2. Add the same difference to 25.

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Example #1:

$$54^2 = ?$$

Step 1. Get the difference between the given number and 50 and square it.

$$54 - 50 = 4; 4^2 = 16$$

Step 2. Add the same difference to 25.

$$25 + 4 = 29;$$

Answer:

$$54^2 = 2916$$

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Example #2:

$$53^2 = ?$$

Step 1. Get the difference between the given number and 50 and square it.

$$53 - 50 = 3; 3^2 = 9$$

Step 2. Add the same difference to 25.

$$25 + 3 = 28;$$

⭐️ As Step 1 returned a single digit number and we can occupy 2 digits in the end, we will add "0" before the number and occupy 2 spaces in the end

$$3^2 = 09;$$

Answer:

$$53^2 = 2809$$

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Example #3:

$$68^2 = ?$$

Step 1. Get the difference between the given number and 50 and square it.

$$68 - 50 = 18; 18^2 = 324$$

⭐️This is why learning squares till 25 is a prerequisite.

Step 2. Add the same difference to 25.

$$25 + 18 = 43;$$

⭐️ As Step 1 returned a 3-digit number and we can only occupy 2 digits in the end, we will take the third as carry and add it to the answer of Step 2

$$43 + 3 = 46$$

Answer:

$$68^2 = 4624$$

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Example #4

$$67^2 = ?$$

Step 1: Get the difference between the given number and 50 and square it.

$$67 - 50 = 17; 17^2 = 289$$

Step 2: Add the same difference to 25.

$$25 + 17 = 42$$

⭐️ As Step 1 returned a 3-digit number and we can only occupy 2 digits in the end, we will take the third as carry and add it to the answer of Step 2

$$42 + 2 = 44$$

Answer:

$$47^2 = 4489$$

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Square of Numbers with base 100:

Numbers less than 100:

1. Get the difference between the given number and 100 and square it.

2. Subtract the same difference from the same number.

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Example #1:

$$94^2 = ?$$

Step 1. Get the difference between the given number and 100 and square it.

$$100 - 94 = 6; 6^2 = 36$$

Step 2. Subtract the same difference from the same number.

$$94 - 6 = 88;$$

Answer:

$$94^2 = 8836$$

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Example #2:

$$93^2 = ?$$

Step 1. Get the difference between the given number and 100 and square it.

$$100 - 93 = 7; 7^2 = 49$$

Step 2. Subtract the same difference from the same number.

$$93 - 7 = 86;$$

Answer:

$$93^2 = 8649$$

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Example #3:

$$88^2 = ?$$

Step 1. Get the difference between the given number and 100 and square it.

$$100 - 88 = 12; 12^2 = 144$$

Step 2. Subtract the same difference from the same number.

$$88 - 12 = 76;$$

⭐️ As Step 1 returned a 3-digit number and we can only occupy 2 digits in the end, we will take the third as carry and add it to the answer of Step 2

$$76 + 1 = 77$$

Answer:

$$88^2 = 7744$$

⭐️ 7744 is the only 4-digit prime number of the type "xxyy"

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Example #4

$$87^2 = ?$$

Step 1: Get the difference between the given number and 100 and square it.

$$100 - 87 = 13; 13^2 = 169$$

Step 2: Subtract the same difference from the same number.

$$87 - 13 = 74$$

⭐️ As Step 1 returned a 3-digit number and we can only occupy 2 digits in the end, we will take the third as carry and add it to the answer of Step 2

$$74 + 1 = 75$$

Answer:

$$87^2 = 7569$$

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Numbers greater than 100:

1. Get the difference between the given number and 100 and square it.

2. Add the same difference to the same number.

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Example #1:

$$104^2 = ?$$

Step 1. Get the difference between the given number and 100 and square it.

$$104 - 100 = 4; 4^2 = 16$$

Step 2. Add the same difference to the same number.

$$104 + 4 = 108;$$

Answer:

$$104^2 = 10816$$

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Example #2:

$$103^2 = ?$$

Step 1. Get the difference between the given number and 100 and square it.

$$103 - 100 = 3; 3^2 = 9$$

Step 2. Add the same difference to the same number.

$$103 + 3 = 106;$$

⭐️ As Step 1 returned a single digit number and we can occupy 2 digits in the end, we will add "0" before the number and occupy 2 spaces in the end

$$3^2 = 09;$$

Answer:

$$103^2 = 10609$$

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Example #3:

$$118^2 = ?$$

Step 1. Get the difference between the given number and 100 and square it.

$$118 - 100 = 18; 18^2 = 324$$

⭐️This is why learning squares till 25 is a prerequisite.

Step 2. Add the same difference to the same number.

$$118 + 18 = 136;$$

⭐️ As Step 1 returned a 3-digit number and we can only occupy 2 digits in the end, we will take the third as carry and add it to the answer of Step 2

$$136 + 3 = 139$$

Answer:

$$118^2 = 13924$$

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Example #4

$$117^2 = ?$$

Step 1: Get the difference between the given number and 100 and square it.

$$117 - 100 = 17; 17^2 = 289$$

Step 2: Add the same difference to the same number.

$$117 + 17 = 134$$

⭐️ As Step 1 returned a 3-digit number and we can only occupy 2 digits in the end, we will take the third as carry and add it to the answer of Step 2

$$134 + 2 = 136$$

Answer:

$$117^2 = 13689$$

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Square of Numbers with base 200:

Numbers less than 200:

1. Get the difference between the given number and 200 and square it.

2. Subtract the same difference from the same number and then multiply by 2.

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Example #1:

$$194^2 = ?$$

Step 1. Get the difference between the given number and 200 and square it.

$$200 - 194 = 6; 6^2 = 36$$

Step 2. Subtract the same difference from the same number and then multiply it by 2.

$$194 - 6 = 188$$

$$ 188 * 2 = 376$$

Answer:

$$194^2 = 37636$$

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Example #2:

$$193^2 = ?$$

Step 1. Get the difference between the given number and 200 and square it.

$$100 - 93 = 7; 7^2 = 49$$

Step 2. Subtract the same difference from the same number and then multiply it by 2.

$$193 - 7 = 186;$$

$$ 186 * 2 = 372$$

Answer:

$$193^2 = 37249$$

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Example #3:

$$188^2 = ?$$

Step 1. Get the difference between the given number and 200 and square it.

$$200 - 188 = 12; 12^2 = 144$$

Step 2. Subtract the same difference from the same number and then multiply it by 2.

$$188 - 12 = 176;$$

$$ 176 * 2 = 352$$

⭐️ As Step 1 returned a 3-digit number and we can only occupy 2 digits in the end, we will take the third as carry and add it to the answer of Step 2

$$352 + 1 = 353$$

Answer:

$$188^2 = 35344$$

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Example #4

$$187^2 = ?$$

Step 1: Get the difference between the given number and 200 and square it.

$$200 - 187 = 13; 13^2 = 169$$

Step 2: Subtract the same difference from the same number and then multiply it by 2.

$$187 - 13 = 174$$

$$ 174 * 2 = 348$$

⭐️ As Step 1 returned a 3-digit number and we can only occupy 2 digits in the end, we will take the third as carry and add it to the answer of Step 2

$$348 + 1 = 349$$

Answer:

$$187^2 = 34969$$

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Numbers greater than 200:

1. Get the difference between the given number and 200 and square it.

2. Add the same difference to the same number and then multiply by 2.

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Example #1:

$$204^2 = ?$$

Step 1. Get the difference between the given number and 200 and square it.

$$204 - 200 = 4; 4^2 = 16$$

Step 2. Add the same difference to the same number and then multiply by 2.

$$204 + 4 = 208;$$

$$ 208*2=416$$

Answer:

$$204^2 = 41616$$

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Example #2:

$$203^2 = ?$$

Step 1. Get the difference between the given number and 200 and square it.

$$203 - 200 = 3; 3^2 = 9$$

Step 2. Add the same difference to the same number.

$$203 + 3 = 106;$$

$$ 206*2=412$$

⭐️ As Step 1 returned a single digit number and we can occupy 2 digits in the end, we will add "0" before the number and occupy 2 spaces in the end

$$3^2 = 09;$$

Answer:

$$203^2 = 41209$$

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Example #3:

$$218^2 = ?$$

Step 1. Get the difference between the given number and 200 and square it.

$$218 - 200 = 18; 18^2 = 324$$

Step 2. Add the same difference to the same number and then multiply by 2.

$$218 + 18 = 236;$$

$$ 236 * 2 = 472$$

⭐️ As Step 1 returned a 3-digit number and we can only occupy 2 digits in the end, we will take the third as carry and add it to the answer of Step 2

$$472 + 3 = 475$$

Answer:

$$218^2 = 47524$$

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Example #4

$$217^2 = ?$$

Step 1: Get the difference between the given number and 200 and square it.

$$217 - 200 = 17; 17^2 = 289$$

Step 2: Add the same difference to the same number and then multiply by 2.

$$217 + 17 = 234$$

$$ 234 * 2 = 468$$

⭐️ As Step 1 returned a 3-digit number and we can only occupy 2 digits in the end, we will take the third as carry and add it to the answer of Step 2

$$468 + 2 = 470$$

Answer:

$$217^2 = 47089$$

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The general formula for Square of Numbers:

Step 1:
Select a Base (numbers like 50, 100, 150, 200, 250, 300...)
For example: The base for 123 should be 100 and for 145 should be 150

Step 2:
Get the difference between the Base and the Number and Square it.
For example: if your number is 311, your base would be 300; so the difference between them is 11 and its square is 121

Step 3:
Case 1: If the number is smaller than your base:
Subtract the difference from the given number and multiply it by the factor Base/100
For example: If the given number is 288, the Base would be 300
The difference between the Base and Number is 300 - 288 =12;
The Square of the difference is 144;
Subtracting the difference from the given number, 288 -12 = 276;
Multiplying this number by the factor of Base/100 i.e., 300/100 = 3, 276 X 3 = 752;
As we can only assign the last two digits of the square of the difference, 1 will be carried to this number, i.e., 752 + 1 = 753;
The final answer comes out to be 75344.

Case 2: If the number is greater than your base:
Add the difference from the given number and multiply it by the factor Base/100

For example: If the given number is 207, the base would be 200
The difference between the Base and Number is 207 - 200 = 7;
The Square of the difference is 49;
Adding the difference from the given number, 207 +7 = 214;
Multiplying this number by the factor of Base/100 i.e., 200/100 = 2, 214 X 2 = 428;

The final answer comes out to be 42849.

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Key points to remember:

• By-heart the Squares of numbers till 25, it will enable you to easily go through this trick.

• Choose your base carefully, and ensure that the given number is in the proximity of 25 numbers of the base.
  For example:
  The base for 124 should be 100,
  The base for 126 should be 150,
  The base for 125 could be both, 100 and 150 as the difference is the same in both cases.

• Make sure you do not forget to multiply by the factor of Base/100 before arriving at the final answer.

• Practice this technique as many times as possible.

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I hope this helps. Please practice this before putting it to practice.

Happy Learning!!