To square a number means to raise it to the second power- that is, to use the number as a factor twice. Finding square root is an opposite operation of squaring.

A square root of a number is one of its two equal factors. Thus $3$ is a square root of $9$ because $3 \times 3$ = $9$. likewise, $-3$ is also a square root of 9 because $(-3) \times (-3) = 9$. In general , $a$ is a square root of $b$ if $a^{2}$ = $b$.
The symbol $\sqrt{}$ called the radical sign which is used to designate square root. The number under the radical sign is called the radicand.
The entire expression such as $\sqrt{9}$ is called a radical. $\sqrt [n]{a}$ denotes the nth root of given number $a$.

The division property of radicals is given as

If $a \geq 0$ and $b > 0$, then

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

That is the square root of the quotient of two positive numbers is the quotient of their square roots.

Example:
Solve the radical $\sqrt{\frac{100}{25}}$ = $\frac{\sqrt{100}}{\sqrt{25}}$

but $\sqrt{100} = 10$ and $\sqrt{25} = 5$

$\therefore$ $\sqrt{\frac{100}{25}}$ = $\frac{10}{5}$ = $2$
Here we will learn how to simplify radicals that contain variables. Every radical expression with variables in the radicand needs to be analysed individually to determine the necessary restrictions on the variable. Consider the radical $\sqrt{x^{2}}$ for different values of $x$.

Let $x = 3$; then $\sqrt{x^{2}}$ = $\sqrt{3^{2}}$ = $\sqrt{9}$ = $3$

Let $x = -3$; then $\sqrt{x^{2}}$ = $\sqrt{(-3)^{2}}$ = $\sqrt{9}$ = $3$.

Thus if $x \geq 0$, then $\sqrt{x^{2}}$ = $x$, but if $x < 0$, then $\sqrt{x^{2}}$ = $-x$. Using the concept of absolute value, we can state for all the real numbers, $\sqrt{x^{2}}$ = $|x|$.

Example: Simplify $\sqrt{72x^{2}y^{7}}$

$\sqrt{72x^{2}y^{7}}$ = $\sqrt{36x^{2}y^{6}}$$\sqrt{2xy}$

= $6xy^{3}\sqrt{2xy}$
It is easier to solve the expression with radicals and exponents. To do this, remember that the numerator of the rational exponent is the power, and the denominator is the index of the radical:

$x^{\frac{m}{n}}$ = $\sqrt[n]{x^{m}}$ = $(\sqrt[n]{x})^{m}$.

Example: We can rewrite 3$^{\frac{2}{5}}$ = $\sqrt[5]{3^{2}}$ = $(\sqrt[5]{3})^{2}$
The following are the example for radicals simplification.

Solved Examples

Question 1: Simplify $\frac{3}{\sqrt[3]{4x}}$.
Solution:
 
$\frac{3}{\sqrt[3]{4x}}$ = $\frac{3}{\sqrt[3]{4x}}$. $\frac{\sqrt[3]{2x^{2}}}{\sqrt[3]{2x^{2}}}$

= $\frac{3\sqrt[3]{2x^{2}}}{\sqrt[3]{8x^{3}}}$

= $\frac{3\sqrt[3]{2x^{2}}}{2x}$

 

Question 2: Solve $\sqrt{\frac{400x^{3}y}{4xy^{5}}}$.
Solution:
 
$\sqrt{\frac{400x^{3}y}{4xy^{5}}}$ = $\sqrt{\frac{100x^{2}}{y^{4}}}$

= $\frac{\sqrt{100x^{2}}}{\sqrt{y^{4}}}$

= $\frac{10x}{y^{2}}$