In square root of numbers, the numbers are multiplied with the same number to get a resultant number. The square root ‘n' of a number ‘x' is that number which when multiplied by itself gives the given number ‘x' as the product. i.e $\sqrt{x}$ times $\sqrt{x}$

(i) We use the sign v to indicate the square root of a number.
i.e., $\sqrt{81}$ = 9, $\sqrt{225}$ = 15,etc.
(ii) We can calculate the square root of positive numbers only. However, the square root of a positive number may be a positive integer, e.g., $\sqrt{25}$ = + 5
The square of 2 is 4, then we can also say that the square root of 4 is 2
If, 6 × 6 = 36 = 62, then square root of 36 is 6
And 8 × 8 = 64= 82, then square root of 64 is 8
Thus, n is the square root of a number x, if x = n × n= n2

The square root of a number is that number which when multiplied by itself gives the given number as the product.

We denote the square root of a number by $\sqrt{}$

Example $\sqrt{x}$ , $\sqrt{2}$ , $\sqrt{9}$

The expression of the square exponents is nothing but putting this in the form of n1/2
The square of this exponent would give us back the original number.
( n1/2 )2 = n2/2 =n
The expression of n1/2 often written as $\sqrt{n}$ and is called the square root of ‘n’.

The symbol of root over is better known as radical.

Example for square exponents:

(81/3 ) = $\sqrt[3]{8}$ = $\sqrt[3]{2\times2\times2}$ = 2

Radicals include square roots v, but a radical can be a cube root or some other root such as $\sqrt[5]{2}$ . Finding square root is the universal operation of squaring or we could also say that squaring can undo a square root. In the radical 3$\sqrt{2}$ , 3 is considered as a coefficient and 2 as the radical and, where 3 is multiplied by $\sqrt{2}$.

The Four Rules of Radicals are :
(i) m = $\sqrt[n]{x}$ if both m = 0 and mn = x
Example: $\sqrt[3]{27}$ = 3

(ii) if ‘n’ is odd then $\sqrt[n]{m^n}$ = m
Example: $\sqrt[3]{(-5)^3}$ = -5

(iii) if ‘n’ is even then $\sqrt[n]{m^n}$ = ImI
Example: $\sqrt[3]{(-5)^3}$ = I-5I = 5

(iv) if m = 0, then $\sqrt[n]{m^n}$= m
Example: $\sqrt[5]{\pi^5}$ = $\pi$

Multiplying square roots of numbers involve the following rules:

Rule 1: The product of the square root of two or more non negative numbers will be equal to the square root of their product. Example: $\sqrt{3}$ $\sqrt{5}$ $\sqrt{6}$ = $\sqrt{90}$ = 3$\sqrt{10}$

Rule 2: The square of the square root of a number equals the number and so when we are squaring the square root of a number we need to just eliminate the radical sign.
Example: $(\sqrt{6})^2$ = ($\sqrt{6}$) ($\sqrt{6}$) = 6

To multiply square root monomials we have to follow a set of rules as they are:

(a) Multiply coefficients and radicals separately
(b) Multiply the resultant products
(c) If required we have to simplify

Example of square root of monomial:

4$\sqrt{3}$ . 2$\sqrt{6}$ = 4.2$\sqrt{3}$ $\sqrt{6}$ = 8$\sqrt{18}$ = 8(3$\sqrt{2}$) = 24$\sqrt{2}$

Dividing square roots involve the quotient rule for square roots and it is found that square roots of the quotient of two numbers are equal to the quotient of their square roots.

For any real numbers ‘a’ and ‘b’

We have $\sqrt{\frac{a}{b}}$ = $\frac{\sqrt{a}}{\sqrt{b}}$

Example for dividing square roots:

$\sqrt{\frac{81}{16}}$ = $\frac{\sqrt{81}}{\sqrt{16}}$ = $\frac{9}{4}$ = 2 $\frac{1}{4}$

Negative numbers have square roots and at the same time their square roots are not real numbers.
They do not have any relevance on the real number line.
For example, the expression $\sqrt{-4}$ does not represent a real number as there is no real number that could be squared to give -4.
When we are dealing with negative numbers and radicals we have to be careful about the restrictions of putting negative numbers under even roots.

We could put negative signs in front of radicals and negative numbers under odd roots and still obtain real numbers.

Examples of Square Roots of Negative Numbers:

The $\sqrt{-4}$ is not a real number since there is no real number whose square is -4.
The (-$\sqrt{25}$) = -5 is the negative square root of 25

As every positive real number ‘r’ has two square roots (vr and -vr) every negative number has two square roots as well.
If ‘-r’ is a negative number then its square roots are ± I $\sqrt{r}$, because (I $\sqrt{r}$)2 = i2r = -r
So in other words, if ‘–r’ is negative, then the principal square root of ‘–r’ is ‘ $\sqrt{-r}$ = i $\sqrt{r}$ ’
The square roots of ‘-r’ would be ‘i $\sqrt{r}$’ and ‘-i $\sqrt{r}$ ’
To avoid confusion, we usually write it as ‘i $\sqrt{b}$’ instead of ‘$\sqrt{b}$ i’

Square root of a perfect square by the Prime Factorization method

When a given number is a perfect square.
There are three steps of finding square roots using the factorization method:

Step 1 : Resolve the given number into a prime factor.
Step 2 : Make pairs of similar factors.
Step 3 : Take the product of the prime factors, choosing one factor out of every pair.

Examples on Square roots with the Factorization method

1. Square root of 2025 or $\sqrt{2025}$
Now, least common multiple (L.C.M) of 2025=3×3×3×3×5×5.
By prime factorization, we get
[2025 = 3×3×3×3×5×5]
= $\sqrt{(3\times3\times3\times3\times5\times5)}$ [make pairs of similar factor]
= 3×3×5
= 45

2. Square root of 1764 or $\sqrt{1764}$
Now, the least common multiple (L.C.M) of 1764 = 2×2×3×3×7×7
By prime factorization, we get
[1764 = 2×2×3×3×7×7]
= $\sqrt{(2\times2\times3\times3\times7\times7)}$ [make pairs of similar factor]
= 2×3×7
= 42

3. Square root of 4356 or $\sqrt{4356}$
Now, we find the least common multiple (L.C.M) of 4356 = 2×2×3×3×11×11.
By prime factorization, we get
[4356 = 2×2×3×3×11×11 ]
= $\sqrt{(2\times2\times3\times3\times11\times11)}$ [make pairs of similar factors]
= 2×3×11
= 66

4. Square root of 11025 or $\sqrt{11025}$
Now, least common multiple (L.C.M) of 11025 = 3×3×5×5×7×7
By prime factorization, we get
[11025 = 3×3×5×5×7×7]
= $\sqrt{(3\times3\times5\times5\times7\times7)}$ [make pairs of similar factors]
= 3×5×7
= 105

Simplified form is not always the least complicated expression. In many cases the simplified expression looks more complicated than the original form or expression.
The radical expression is considered to be in simplified form if

(a) There are no perfect squares that are factors of the quantity under the square root sign,
(b) There are no fractions under the sign of the radical.
(c) There are no radicals present in the denominator or the denominator is rationalized.

Examples of removal of perfect nth powers from the radical:

$\sqrt[3]{48}$ = $\sqrt[3]{2^3 (6)}$ = $\sqrt[3]{2^3}$ . $\sqrt[3]{(6)}$ = 2 $\sqrt{6}$

Example of reduction of radical index:

$\sqrt[4]{64}$ = $\sqrt[4]{2^6}$ = 26/4 = 23/2 = $\sqrt{8}$ = 2$\sqrt{2}$

Example of rationalizing the denominator:

$\sqrt[3]{\frac{9}{2}}$ = $\sqrt[3]{\frac{9 (2)^2}{2^3}}$ = $\frac{\sqrt[3]{36}}{2}$

Some of the examples where we can observe similar pairs:

1. $\sqrt{9}$
= $\sqrt{3\times3}$ [here square root of 9 is 3×3 that means 3 is in pair]
= $\sqrt{3^2}$ [here square roots of 32 ]
= 3

2. $\sqrt{12}$
= $\sqrt{2\times2\times3}$ [here 2×2 is in pair so 2 will go outside the square root]
= 2 $\sqrt{3}$ [here 2 is outside the square root and 3 is inside the square root]

3. $\sqrt{18}$
= $\sqrt{2\times3\times3}$ [here 3×3 is in pair so 3 will go outside the square root]
= 3 $\sqrt{2}$ [here 3 is outside the square root and 2 is inside the square root]

4. $\sqrt{49}$
= $\sqrt{7\times7}$ [here 7×7 is in pair so 7 will go outside the square root]
= 7

5. $\sqrt{324}$
= $\sqrt{2\times2\times3\times3\times3\times3}$ [here we can observe that 2×2×3×3×3×3 ,that means 2, 3, 3 numbers are in pairs so one number from each pair will go outside the square root]
= 2×3×3
= 18

6. $\sqrt{720}$
= $\sqrt{2\times2\times2\times2\times3\times3\times5}$ [here similarly 2×2×2×2×3×3×5 and we can observe that 2, 2, 3 are in pairs so one number from each pair will go outside the square root and we can also see that 5 is not in pairs so 5 will be inside the square root]
= 2×2×3 $\sqrt{5}$
= 12 $\sqrt{5}$

7. Find the square root of the perfect square 5184.

5184 = 2×2×2×2×2×2×3×3×3×3

[First we find the prime factors of the given perfect square. Since the number is a perfect square, therefore, we get pairs of similar prime factors].

$\sqrt{5184}$ =2×2×2×3×3 [here choose one factor from each pair and multiply together].

= 72 [the result will be the square root of the given number].

8. Find the value of $\sqrt{\frac{25}{64}}$.

$\sqrt{\frac{25}{64}}$ = $\frac{\sqrt{25}}{\sqrt{64}}$ [here both are equal].

= $\frac{\sqrt{5\times5}}{\sqrt{8\times8}}$ [here first we find the prime factors of the numerator and the denominator. Since the number is a perfect square, therefore, we get pairs of similar prime factors].

= $\frac{5}{8}$ [here one factor is chosen from each pair]

9. $\sqrt{18}$

= $\sqrt{2\times3\times3}$ [here 3×3 is in pair so 3 will go outside the square root]

= 3 $\sqrt{2}$ [here 3 is outside the square root and 2 is inside the square root]

10. Find the smallest number by which 252 must be multiplied so that the product becomes a perfect square.

252 = 2×2×3×3×7

We see that, in prime factorization of 252, there exists a number 7 which is unpaired.

Hence , to make a pair of 7, we will have to multiply 252 by 7

Hence, 7 is the required number.

So, 252 × 7 = 1764 now you can notice that 2×2×3×3×7×7 [is a perfect square] = 2×3×7 = 42.