Let’s see how we can find square of a number faster using Vedic Mathematics (Nikhilam method)

__Step 1__

$10$ is the nearest power of $10$ which can be taken as our base. The deviation to our base $=12-10=2$ (To find the deviation, just remove the leftmost digit "$1$" and you will get it quickly)

Left side of the answer is the sum of the number and deviation. Hence, left side of the answer $=12+2=14$

__Step 2__

Our base $10$ has a single zero. Therefore, right side of the answer has a single digit and that can be obtained by taking the square of the deviation.

Hence, right side of the answer $=2^2=4$

Therefore, answer $=144$

__Step 1__

$10$ is the nearest power of $10$ which can be taken as our base. The deviation to our base $=13-10=3$ (To find the deviation, just remove the leftmost digit "$1$" and you will get it quickly)

Left side of the answer is the sum of the number and deviation. Hence, left side of the answer $=13+3=16$

__Step 2__

Our base $10$ has a single zero. Therefore, right side of the answer has a single digit and that can be obtained by taking the square of the deviation.

Hence, right side of the answer $=3^2=9$

Therefore, answer $=169$

__Step 1__

$10$ is the nearest power of $10$ which can be taken as our base. The deviation to our base $=14-10=4$ (To find the deviation, just remove the leftmost digit "$1$" and you will get it quickly)

Left side of the answer is the sum of the number and deviation. Hence, left side of the answer $=14+4=18$

__Step 2__

Our base $10$ has a single zero. Therefore, right side of the answer has a single digit and that can be obtained by taking the square of the deviation.

Hence, right side of the answer $=4^2=16.$ But right side of the answer can have only a single digit because our base $10$ has a only single zero. Hence, from the obtained number $16,$ we will take right side as $6$ and $1$ is taken as a carry which will be added to our left side. Hence left side becomes $18+1=19$

Therefore, answer $=196$

__Step 1__

$100$ is the nearest power of $10$ which can be taken as our base. The deviation to our base $=106-100=6$ (To find the deviation, just remove the leftmost digit "$1$" and you will get it quickly)

Left side of the answer is the sum of the number and deviation. Hence, left side of the answer $=106+6=112$

__Step 2__

Our base $100$ has two zeros. Therefore, right side of the answer has two digits and that can be obtained by taking the square of the deviation.

Hence, right side of the answer $=6^2=36$

Therefore, answer $=11236$

__Step 1__

$100$ is the nearest power of $10$ which can be taken as our base. The deviation to our base $=112-100=12$ (To find the deviation, just remove the leftmost digit "$1$" and you will get it quickly)

Left side of the answer is the sum of the number and deviation. Hence, left side of the answer $=112+12=124$

__Step 2__

Our base $100$ has two zeros. Therefore, right side of the answer has two digits and that can be obtained by taking the square of the deviation.

Hence, right side of the answer $=12^2=144.$ But right side of the answer can have only two digits because our base $100$ has only two zeros. Hence, from the obtained number $144,$ we will take right side as $44$ and $1$ is taken as a carry which will be added to our left side. Hence left side becomes $124+1=125$

Therefore, answer $=12544$

**Note**

This method is extremely useful for competitive examinations and if practiced well, square of a number can be determined within seconds using the same.

suppose we want to find the square of $96$

$100-96=4$

$96-4/4×4\\

9216 $

$96^2=9216$

lets try another number

$87^2\\

100-87=13\\

87-13/13×13\\

74/169$

$74+1/69\\

7569$

Now try to square of $46$

$50-4=46$

$\dfrac{46-4}{2}/4×4\\

21/16\\

2116$

any brilliant here?

Lets take a example of $2.5$

By multiplying it with $100$ it becomes $250$

Now think of the nearest perfect square and that is in this case is $256$

Now find the difference b/w that perfect square and the given no.

the difference b/w $256$ and $250$ is $6$

Now think of the very basic expansion equation $(a-b)^2=a^2-2ab+b^2$

so now in this case, we can say that

$250=(16-x)^2=256-2×16×x+x^2$

Here $x$ is very small. So we can neglect it so.

$6=2×16×x$

So $x=\dfrac{6}{32}$ which is approx $0.1875$

So subtract it from $16$ that makes it $15.8125$

Now remind that we have multiplied the original no. with $100.$ So after rooting we have to divide the answer with $10$

so final answer is $1.58125$ approx.

hope this helps.

for eg $2.5$

square of $25=625$

now in $2.5$ decimal is preceding $1$ digit

so in $625$ answer will precede $2$ digits. So final answer will be $6.25$

Want answer with above method in steps.

If given it will be helpful for me

$39\qquad \qquad-1\\

39\qquad \qquad -1$

-----------------------

$(39-1)~/~(-1×-1)$

$~~38~\qquad /\qquad 1$

Since it is to base $40,$ multiply only the $38$ part with $4$ $=152$

$(38×4)/1$

$152 / 1$

Ans $=1521$

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