2.Square and Square root of a Number -General Aptitude

       Square and Square root of a Number 

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

Example 1: Find square of $12$

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 $=144$

Example 2: Find square of $13$

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 $=169$

Example 3: Find square of $14$

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 $=196$

Example 4: Find square of $106$

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 $=11236$

Example 5: Find square of $112$

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 $=12544$

Shortcut method to find the square of any two digit number:

suppose we want to find the square of $96$ 

$100-96=4$

$96-4/4\times 49216$

${96}^2=9216$

lets try another number 

${87}^{2}100-87=1387-13/13\times 1374/169$

$74+1/697569$

Now try to square of $46$ 

$50-4=46$

$\frac{46-4}{2}/4\times 421/162116$

$\mathrm{How\; can\; we\; calculate\; square\; of 97?Take\; base 10010097-100=-397-100=-3 i.e.,\; deviation\; is -3-3Left\; side =97+(-3)=94=97+(-3)=94right\; side =(-3)2=9=(-3)2=9Since\; base\; has\; two\; zeros,\; write\; it\; as 0909Hence,\; answer\; is 9409}$$97$

$2.5,44.5?$

how to find square roots in decimal places like $2.5$$,$$44.5$$?$$2.5,44.5?$

First of all multiply the number with $100$ and keep that in mind.

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
$250=(16-x{)}^2=256-2\times 16\times x+x^2$

Here $x$ is very small. So we can neglect it so.
$6=2\times 16\times x$
So $x=\frac{6}{32}$ 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.

392=?

Consider the base as $40$$40$

$39-139-1$
-----------------------
$(39-1)/(-1\times -1)$
$38/1$   

Since it is to base $40,$ multiply only the $38$ part with $4$ $=152$
$(38\times 4)/1$
$152/1$

Ans $=1521$

Square Root

The square root of a number is that number which when multiplied by itself gives the given number. It is denoted by the symbol √.

What is the square root of a number x?

The square root of a number x is that number which when multiplied by itself gives x as the product. We denote the square root of a number x by √x. 

For example:

(i) 1² = 1

Therefore, square root of 1 is 1. Symbolically √1 = 1.

(ii) 2² = 4

Therefore, square root of 4 is 2. Symbolically √4 = 2.

(iii) 3² = 9

Therefore, square root of 9 is 3. Symbolically √9 = 3. 

(iv) 5 is the square root of 25. Since 5² = 25

Or we can write it as √25 = 5 (Square root of 25 is 5) 

(v) ²/₃ is the square root of ⁴/₉. Since 2²/3² = ⁴/₉

Or we can write it as √(⁴/₉) = ²/₃ (Square root of ⁴/₉ is ²/₃)

(vi) 0.2 is the square root of 0.04. Since 0.2² = 0.04

Or we can write it as √0.04 = 0.2 (Square root of 0.04 is 0.2) 

In general; if n = m², then m is the square root of n, i.e., m = √n

Unit Digits of Squares

First we need to remember unit digits of all squares from 1 to 10. The figure below shows the unit digits of the squares.

Now from the picture we can say that, whenever the unit digit of a number is 9, unit digit of the square root of that number will be definitely 3 or 7. Similarly, this can be applied to other numbers with different unit digits.

How to Find Square Root of a Four Digit Number

Let’s learn how to find square root by taking different examples.

Example: Find the square root of 4489.

We group the last pair of digits, and the rest of the digits together.

Now, since the unit digit of 4489 is 9. So we can say that unit digit of its square root will be either 3 or 7.

Now consider first two digits i.e. 44. Since 44 comes in between the squares of 6 and 7 (i.e. 62 < 44 < 72), so we can definitely say that the ten’s digit of the square root of 4489 will be 6.  So far, we can say that the square root will be either 63 or 67.

Now we will find the exact unit digit.

To find the exact unit digit, we consider the ten’s digit i.e. 6 and the next term i.e. 7.

Multiply these two terms

Since, 44 is greater than 42. So square root of 4489 will be the bigger of the two options i.e. 67.
Let us take another example.

Example: What is the square root of 7056?

Unit digit will be 4 or 6.

Since, 82 < 70 < 92

So the square root will be either 84 or 86.

Now consider 8 and 9

Since, 70 is less than 72. So square root will be the lesser of the two values i.e. 84.

Let’s try it out with five digit numbers now!

How to Find Square Root of a Five Digit Number

We pair the digits up starting from the right side. Since there is one extra left over after two pairs are formed, we club it with the pair closest to it.

Example: √(16641) = ?

Unit digit will be 1 or 9.

Since, 122 < 166 < 132

So, the square root will definitely be 121 or 129.

Now, consider 12 and 13

Since, 166 is greater the 156, we pick the larger of the options i.e. 129.

 Let’s take another example, so that this trick will be clear to you.

Example: √(33489) = ?

Unit digit will be 3 or 7.

Since, 182 < 334 < 192

So, square root will be 183 or 187.

Now consider 18 and 19.

Now, 334 is less than 342. So, the square root will be lesser of the two numbers i.e. 183.

This trick to find square root of a number will surely help you in your exams

PRACTICE SOME QUESTIONS AND BE PERFECT.

ALL THE BEST!!!!!!!!!

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