97² = 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) : 3² = 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): 
2² = 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 × aⁿ = a^m+n

a^{m ÷ n} = a^{m - n}

(a^m)ⁿ = a^mn

a^{($\frac{1}{m}$)} = $\sqrt[m]{a}$

a ^-m = a^{$\frac{1}{m}$}

a^{($\frac{m}{n}$)} = $$\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.