Top

# Square Root

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, " $\sqrt{}$ "

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).
Square Root Definition:

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, $\sqrt{3} \times \sqrt{3}$ = $3$.
Perfect Square Factors:

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 $100$.
Properties of Square roots for simplifying Square Roots expressions:

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

They are as follows:
1) Product Property: The Square Roots product property is nothing but the product of two different or same value square roots which is equivalent to the square root of their product.

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

2) Quotient Property: The Square Roots quotient property is nothing but the quotient of any two different or same value square roots equal to the square root of their quotient.
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 $\sqrt{\frac{a}{b}}$ = $\frac{\sqrt{a}}{ \sqrt{b}}$.

Square root properties:

Solved Examples:

Example 1:

Find the square root of 121?

Solution:

$x^2$ = $121$

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

$x$ =$\pm 11$

Example 2:

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

Solution:

Example 3:

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

Solution:

 Related Calculators Add Square Roots Calculator calculate root mean square Calculate Square Root Fraction Square Root Calculator

 More topics in  Square Root Simplifying Square Roots
*AP and SAT are registered trademarks of the College Board.