 # Square Root

Square root definition can be defined as a number which when multiplied by itself gives a as the product, then it is known as a square root. For example, since

5 × 5 = 25, so $\sqrt{25}$ = 5

All real numbers have two square roots, one is a positive square root and another one is a  negative square root. The positive square root is also referred to as the principal square root. Two positive numbers when multiplied results in a positive number because of having the same sign. As far as negative numbers are concerned, they don't have real square roots because a square is either positive or 0. Square root symbol is represented as  $\sqrt{}$

### Perfect Squares

When two equal numbers are multiplied, it results in a perfect square. For example, 55=25

This is a basic property of the square root

## Square Root Table

 No Value No Value No Value 1 1.000 21 4.583 41 6.403 2 1.414 22 4.690 42 6.481 3 1.732 23 4.796 43 6.557 4 2.000 24 4.899 44 6.633 5 2.236 25 5.000 45 6.708 6 2.449 26 5.099 46 6.782 7 2.646 27 5.196 47 6.856 8 2.828 28 5.292 48 6.928 9 3.000 29 5.385 49 7.000 10 3.162 30 5.477 50 7.071 11 3.317 31 5.568 51 7.141 12 3.464 32 5.657 52 7.211 13 3.606 33 5.745 53 7.280 14 3.742 34 5.831 54 7.348 15 3.873 35 5.916 55 7.416 16 4.000 36 6.000 56 7.483 17 4.123 37 6.083 57 7.550 18 4.243 38 6.164 58 7.616 19 4.359 39 6.245 59 7.681 20 4.472 40 6.325 60 7.746

 61 7.81 81 9 62 7.874 82 9.055 63 7.937 83 9.11 64 8 84 9.165 65 8.062 85 9.22 66 8.124 86 9.274 67 8.185 87 9.327 68 8.246 88 9.381 69 8.307 89 9.434 70 8.367 90 9.487 71 8.426 91 9.539 72 8.485 92 9.592 73 8.544 93 9.644 74 8.602 94 9.695 75 8.66 95 9.747 76 8.718 96 9.798 77 8.775 97 9.849 78 8.832 98 9.899 79 8.888 99 9.95 80 8.944 100 10

Referring to this square root table, half of the energy will be saved.

### How To Find Square Root?

Now, since we know what a square root is, we can quickly jump into knowing how to find a square root of a number. There is no square root formula as such but well, there are two ways to find the square root of a number. They are the Prime Factorization Method and the Division Method. You will know how to solve the square root equation using these two methods.

Finding Square Root Using Prime Factorization Method

To find the square root of a perfect square we have to follow the following steps:

Step 1) First resolve the given number into prime factors.

Step 2) Make pairs of similar factors.

Step 3) The product of prime factors, chosen one out of every pair, gives the square root of the given number.

For example, find the square root of  24336.

Solution) Resolving 24336 into prime factors, we get:

 2 24336 2 12168 2 6084 2 3042 3 1521 3 507 13 169 13 13 1

24336 = $\underline{2}$ × $\underline{2}$ × $\underline{2}$ ×  $\underline{2}$ × $\underline{3}$ ×  $\underline{3}$  × $\underline{13}$  × $\underline{13}$

$\sqrt{24336}$ = 2 × 2 × 3 × 13 = 156

Finding Square Root Using Division Method

In case of large numbers; we find the square root by using the following steps:

Step 1) Mark off the digits in pairs starting with the unit digit. Every pair and remaining one digit (if any) is called a period.

Step 2) Think of the largest number whose square is equal to or just less than the first period. Take this number as the divisor as well as quotient.

Step 3) In this step subtract the product of divisor and quotient from the first period and bring down the next period to the right of the remainder. This becomes the new dividend.

Step 4) now, a new divisor is obtained by taking twice the quotient and annexing with it a suitable digit which is also taken as the next digit of the quotient, chosen in such a way that the product of new divisor and this digit is equal to or just less than the new dividend.

### Solved Examples

Question 1) By using the table of square roots, find the values of

i) $\sqrt{13}$

ii) $\sqrt{83}$

iii) $\sqrt{150}$

iv) $\sqrt{36}$

Solution 1)

i) Let x = 13

From the table, we find that when  x = 13, then $\sqrt{x}$ 3.606.

Therefore, $\sqrt{13}$ = 3.606.

ii) Let x = 83

From the table, we find that when  x = 83, then $\sqrt{x}$ = 9.110.

Therefore,  $\sqrt{83}$=9.110.

iii) $\sqrt{150}$ =  $\sqrt{5 × 5 × 6}$ = 5 × $\sqrt{6}$ = 5 × 2.449

Using the table for $\sqrt{6}$,  = 12.245

Therefore, $\sqrt{150}$ =  12.245

iv)$\sqrt{368}$ = $\sqrt{4 × 4 × 23}$ = 4 × $\sqrt{23}$ = 4 × 4.796

Using the table for $\sqrt{23}$, = 19.184

Therefore, $\sqrt{368}$ = 19.184

Question 2) find the square root of  1764

Solution 2)

Resolving 1764 into prime factors, we get:

 2 1764 2 882 3 441 3 147 7 49 7 7 1

1764 = $\underline{2}$ × $\underline{2}$ × $\underline{3}$ × $\underline{3}$ × $\underline{7}$ × $\underline{7}$

$\sqrt{1764}$ = 2 × 3 × 7 = 42

Question 3) Find the smallest number which when multiplied by 720 gives a perfect square number.

i) Give the perfect square number so obtained?

ii) Find the square root of this perfect square number.

Solution 3) Resolving 720 into prime factors, we get:

720 = 2 × 2  ×  2  ×  2  ×  3  ×  3  ×  5.

Thus, 2,2,3 exist in pairs while 5 is alone.

So, we should multiply the given number by 5 to get a perfect square number.

i) Perfect square number so obtained = 720 × 8 = 3600

ii) Now, 3600 = 2  ×  2  ×  2  ×  2  ×  3  ×  3  ×  5 ×  5

Therefore, $\sqrt{3600}$ =   2 × 2  ×  3  ×  5 = 60

1. What is the relation between square numbers and square root?

When a square root is multiplied, it produces a square number. Thus, a square root has an inverse operation than the square number. Let’s take an example, the square root of 36 is 6 because 6 x 6 = 36. Finding the square root of a number can be much trickier than calculating the square number. Therefore, the modern calculator has the square root button to make the calculation easier.

2. What are the applications of square roots?

Square root can be very important in mathematical topics such as quadratic equations, algebra, geometry, and calculus.

3. How to find the square root of a number?

There are two methods to find square root i.e prime factorization method and division method. We can also learn how to find square roots without a calculator as nowadays many calculators have a square root function.