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 $46$
$50-4=46$
$\dfrac{46-4}{2}/4×4\\
21/16\\
2116$
$97-100=-3$
i.e., deviation is $-3$
Left side $=97+(-3)=94$
right side $=(-3)^2=9$
Since base has two zeros, write it as $09$
Hence, answer is $9409$
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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