97^{2} = 9409

**Step 1.** Subtract the number from 100: 100- 97 = 3

**Step 2.** Subtract the number ( from Step 1) from original number : 97-3 =94

**Step 3.** Square the result from Step 1 ( if the result is a single digit put a 0 in front of it) : 3^{2} = 09

**Step 4.** Place the result from Step 3 next to the result from Step 2: 9409

Squaring the number nearest the 100

**Let’s try it with (102) ^{2}**

**Step 1.** Add the number to the ones digit:

102 + 2 = 104

**Step 2.** Square the ones digit number ( if the result is a single digit put a 0 in front of it):

2^{2} = 04

**Step 3- **base is 100 multiplying the step 1 by 100 = 10400

**step 4-** add step 2 and step 3 = 10404

This method is to find the multiplication of the number nearest the 100.

**When one number greater than 100 and other is less than 100**

**Let’s try it with 103 X 98**

Step1-Multiplication of (+3 x –2)= –6

Step2 -Now add (3) to 98 or (–2) to 103= 101

Step 3- base is 100 multiplying the step 2 by 100 = 10100

step 4- add step 1 and step 3 = 10100 – 6 = 10094

This method is to find the multiplication of the number nearest the 100.

**When both number less than 100**

**Let’s try it with 98 X 94**

Step1-Multiplication of (–2 x –6)= +12

Step2 -Now add (–2) to 94 or (–6) to 98= 92

Step 3- base is 100 multiplying the step 2 by 100 = 9200

step 4- add step 1 and step 3 = 9200 + 12 = 9212

This method is to find the multiplication of the number nearest the 100.

**When both number Greater than 100**

**Let’s try it with 104 X 108**

Step 1- multiplication of 4 x 8 = 32

Step 2- cross addition – 104 + 8 or 108 + 4= 112

Step 3- base is 100 multiplying the step 2 by 100 = 11200

step 4- add step 1 and step 3 = 11200 + 32 =** 11232**

This method is to find square of the numbers which has unit’s digit as 5.

Step 1: Multiply ten’s digit with its next number.

Step 2: Find square of unit’s digit. i.e.: Square of 5.

Step 3: Write answers of step 1 and step 2 to together or side by side.

Laws of Indices

a^{m} × a^{n} = a^{m+n}

a^{m} ^{÷ n }= a^{m - n}

(a^{m})^{n }= a^{mn}

a^{(}^{$m1 $) }= $ma $

a^{ -m} = a^{$m1 $}

a^{($nm $) }= $$\sqrt[n]{a^{m}}

a^{o} = 1

DIVISIBILITY RULES There are some specific rules by which we can determine the divisor of the given number. Using these rules you can easily determine a divisor of a given number.

MULTIPLICATION

Multiplying two-digit numbers could be tricky, for this you should follow the steps

**Let’s try it with 19 X 15**

STEP 1 First step is same as conventional method, here we multiply 4 with 6.

STEP 2 This is an interesting step. Now multiply last digit first value and first digit of second value and vice-versa. Then we add outcomes. But we need the last number that is 8 here.

While doing multiplication of a two digit number with another two digit number, we take at least 6 steps. Try yourself. Multiply 62 with 32.

Exams

IIT JEE

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