The real number system has two important types of sets of numbers which are rational and irrational numbers, where an irrational numbers are those which are the result of the square root of a non-perfect square.

Notation:

The symbol used for square root is known as a radical sign as shown here, " $\sqrt{}$ "

The number under which the radical sign is present is known as the radicand. A

An index is a
term which is written on the outside of the radical symbol and its
always fixed in case of a square root and is equal to $2$ always (in the
case of square roots).

A square root of a number is a number $r$ such that $r^2$ = $a$, or in words, a number $r$ whose square (this is the number we get
when we multiply the number by itself) is a. For example, the square
root of $4$ is $2$ because $2^2$ is $4$.

The square of a square root is the number itself $\sqrt{a} \times \sqrt{a}$ = $a$. For example, $\sqrt{3} \times \sqrt{3}$ = $3$.

A number is known as a perfect square if it can be expressed as the product of any other whole number in itself twice.

For example, $25$ is a perfect square as it can be expressed as $5 \times 5$.

When
evaluating (often referred to as taking) the square root of a given
number, find if it has any perfect square factors or not.

The following is the list of perfect squares from $1$ to $100$.

There are two properties which help in the process of simplifying the expressions of square root.

They are as follows:

For all non-negative real numbers a and b, $\sqrt{a} \times \sqrt{b}$ = $\sqrt{ab}$ and $\sqrt{ab}$ = $\sqrt{a}$ x $\sqrt{b}$

For all real numbers $a$ and $b$, which
are non-negative, where $b \neq 0$, $\frac{\sqrt{a}}{ \sqrt{b}}$ = $\sqrt{\frac{a}{b}}$ and
$\sqrt{\frac{a}{b}}$ = $\frac{\sqrt{a}}{ \sqrt{b}}$.

Find the square root of 121?

Solution:

$x^2$ = $121$

$\sqrt{x^{2}}$ = $\pm \sqrt{121}$

$x$ =$\pm 11$

Find the value of $\sqrt{27}$ + $\sqrt{12}$.

Solution:

Using the product property of square roots, find $\sqrt{36}$ and $\sqrt54$.

Solution:

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