**Here we will learn a trick how to square a two digit number**** ending with 5:**

First, we will take 2 digit number then we will take the number greater by 1 of the 1st digit of the 2 digit number that is to be squared. Then we will multiply the number considered with the 1st digit of the number to be squared.Then we put the value in front of number 25.

For eg: If we find the square of **15**, then we the number greater by 1 of the 1st digit of the number to be squared, i.e., 2. Then we multiply the number (2) with the 1st digit of the number to be squared, i.e., 1 and get the value =2. Then we put 25 after the value obtained on multiplication and hence the desired result. __2__25 i.e., 15^{2} = 225.

A chart with few examples are given below

We take a no. to be square in the middle as shown.Then we take a no. greater than the no. to be squared. Similarly, we take a number smaller than the number to be squared. The difference between the greater number and the number to be squared must be equal to the difference between the smaller number and the number to be squared.

We take the first digit of the greater number. Then we break the smaller number into two parts. **For eg:** let us find the square of 39. So, let us take 40 (a number that is greater than 39) and 38 ( number that smaller than 39). Here the difference between the greater number (40) and the number to be squared (39) is 1. Similarly, the difference between the smaller number (38) and the number to be squared (39) is 1 .i.e., the difference is equal which is 1 as shown in fig. Then we multiply them and obtain a value (152). We put zero at the end of the value obtained on multiplication, i.e., 1520. Then we find the square of the difference between the numbers ( in this case it is 1^{2 }= 1) and add it to the value obtained after putting zero. Hence we get the result, i.e., 39^{2 }= 1521.

Two examples are shown below for reference.

__SQUARE AND SQUARE ROOTS USING FORMULA:__

FORMULA: Use (a+b)^{2} = a^{2} + 2ab + b^{2}, i.e., (a/b)^{2} = a^{2}/2ab/b^{2} [we replace ‘+’ sign with ‘/’ ]. Apply this formula to obtain the square of a two digit no.

** Square of (76)^{2}** –

First split the number into two parts. Consider a= 7 and b= 6. Applying the formula given above, a^{2 }+ 2ab + b^{2} OR a^{2}/2ab/b^{2} = 7^{2}/2*7*6/6^{2} = 49/84/36.Then go from the right to the left and note down 6 and then carry 3.Then add 3 with 84 ,i.e.,(3 + 84) = 87 and note down 7 on left of 6 and then carry 8.Then again add 8 with 49 , i.e., (8 + 49) = 57 and note down 57 at left of 7. Hence our result (76)^{2} = 5776

** Square of (55)^{2}** –

First split the number into two parts. Consider a= 5 and b= 5. Applying the formula given above, a^{2 }+ 2ab + b^{2} OR a^{2}/2ab/b^{2} = 5^{2}/2*5*5/5^{2} =25/50/25.Then go from the right to the left and note down 5 and then carry 2.Then add 2 with 50 ,i.e.,(2 + 50) = 52 and note down 2 on left of 5 and carry 5.Then again add 5 with 25 , i.e., (5 + 25) = 30 and note down 30 at left of 2. Hence our result (55)^{2} = 3025

__Square of (122) ^{2}__

First split the number into two parts. Consider a=12 and b=2. Applying the formula given above, a^{2 }+ 2ab + b^{2} OR a^{2 }/2ab / b^{2} = 12^{2}/2*12*2/2^{2} =144/48/4. Then go from the right to the left and note down 4 and then carry 0 .Then add 0 with 48 ,i.e.,(0 + 48) = 48 and note down 8 on left of 4 and then carry 4.Then again add 4 with 144 , i.e., (4 + 144) = 148 and note down 148 at left of 8. Hence our result (122)^{2} = 14884.

__Square of (125) ^{2}__

First split the number into 2 parts. Consider a= 12 and b= 5. Applying the formula given above, a^{2 }+ 2ab + b^{2} OR a^{2}/2ab/b^{2} = 12^{2}/2*12*5/5^{2} =144/120/25. Then go from the right to the left and note down ** 5** and then carry 2.Then add 2 with 120 ,i.e.,(2 + 120) = 122 and note down

**on left of 5 and then carry 12.Then again add 12 with 144 , i.e., (12 + 144) =**

__2__**and note down 156 at the left of 2.Hence our result (125)**

__156__^{2}=

__15625__**SQUARE AND SQUARE ROOT OF A NUMBER ENDING WITH 6:**

**Square of (76) ^{2}**–

First of all note down ** 6**.Then multiply 2 with (7+1), i.e., 2*(7+1)= 16. Now add 1 with 16, i.e., (16+1) = 17 and then note down

**to left of 6 and then carry 1. Again multiply 7 with (7+1), i.e., (7+1) * 7= 56. Then add the previous carry 1 with 56, i.e., (56+1)=57 and note down**

__7__**to the left of 7. Hence, the result of (76)**

__57__^{2}=

__5776.__**Square of (46) ^{2}**–

First of all note down ** 6**.Then multiply 2 with (4+1), i.e., 2*(4+1)= 10. Now add 1 with 10, i.e., (10+1) = 11 and note down

**to left of 6 and carry 1. Again multiply 4 with (4+1), i.e., (4+1) * 4= 20 and add previous carry 1 with 20, i.e., (20+1)=21 and note down**

__1__**to left of 1. So, the result (46)**

__21__^{2}=

__2116.__**SQUARE AND SQUARE ROOT OF 100 BASE:**

**Square of **(98)^{2}–

First substract 98 from 1oo, i.e., (100-98)= 2. Then square the result, i.e., 2^{2}=4 and note down 4. As the squared no. is a single digit number so add an extra zero to left of 4, i.e., 04. Now again substract the difference value (2) from the number, i.e., (98-2)= 96 and note down 96 to left of 04. So, the result (98)^{2}= 9604.

**Square of **(91)^{2}–

First substract 91 from 1oo, i.e., (100-91)= 9. Then square the result, i.e., 9^{2}=81 and note down 81. Now again substract the difference value (9) from the number, i.e., (91-9)= 82 and note down 82 to left of 81. So, the result (91)^{2}= 8281.