The square root of x is an algebraic function. It is denoted by the symbol $\sqrt{x}$. In this article, we will learn its definition, graph, various properties, etc. We will also learn how to find its derivative and integration.

## Square Root Definition

If $B^2=x$, then we say that the number $B$ is a **square root** of $x$. Symbolically, we can write it as

$B=\sqrt{x}$.

Square root Symbol: $\sqrt{}$

** Note:** The square root of $x$ can be written as $x^{\dfrac{1}{2}}$, that is, $\sqrt{x}=x^{\dfrac{1}{2}}$.

## Square Root Example

As $3^2=9$, by the above definition, we can say that $3$ is a square root of $9$, that is, $\sqrt{9}=3$. More examples of square roots are given below:

- As $4^2=16$ we have $\sqrt{16}=4$.
- As $5^2=25$ we have $\sqrt{25}=5$.
- As $6^2=36$ we have $\sqrt{36}=6$.

## Square Root Graph

Let $y=\sqrt{x}$. Squaring both sides, we get that

$y^2=x$

Thus, the graph of square root of $x$ represents a parabola. The graph of root $x$ is given below:

Source: Wikipedia

## Square Root Properties

The properties of square roots are given below:

- The product of the Square roots of two numbers is the same as the square root of the product of those two numbers. That is, $\sqrt{a}\cdot \sqrt{b}=\sqrt{a \cdot b}$.
- If $a, b (\neq 0)$ be two real numbers. Then we have $\dfrac{\sqrt{a}}{\sqrt{b}}=\sqrt{\dfrac{a}{b}}$.
- Square root of a perfect square is always an integer. Perfect square is such a number that is the square of some integer. For example, $\sqrt{100}=10$, so $100$ is a perfect square number.
- The sum of the square root of two numbers may not be equal to the square root of the sum of those numbers. That is, $\sqrt{a+b} \neq \sqrt{a}+\sqrt{b}$.
- The square root of $0$ is $0$.
- The square root of $1$ is $\pm 1$.
- The square root of the negative numebr $-1$ is called the imaginary number, and it is denoted by $i$. Thus, $i=\sqrt{-1}$.

## Square Root Derivative

To find the derivative of root x, we will use the power rule: $\dfrac{d}{dx}(x^n)=nx^{n-1}$.

Note that $\sqrt{x}=x^{\dfrac{1}{2}}$.

By the above power rule,

$\dfrac{d}{dx}(\sqrt{x})=\dfrac{d}{dx}(x^{\dfrac{1}{2}})$

$=\dfrac{1}{2} x^{\frac{1}{2}-1}$

$=\dfrac{1}{2} x^{-\frac{1}{2}}$

$=\dfrac{1}{2\sqrt{x}}$

So the derivative of root x is 1/[2root(x)].

## Square Root Integration

To find the integration of root x, we will use the power rule of integration: $\int x^n dx=\dfrac{x^{n+1}}{n+1}$.

Note that $\sqrt{x}=x^{\dfrac{1}{2}}$.

By the above power rule of integration, the integration of root x is

$\int \sqrt{x} dx=\int x^{\dfrac{1}{2}} dx$

$=\dfrac{x^{\dfrac{1}{2} +1}}{\dfrac{1}{2}+1}$

$=\dfrac{x^{3/2}}{3/2}$

$=\dfrac{2}{3} x^{3/2}$

$=\dfrac{2}{3} x \cdot x^{\dfrac{1}{2}}$

$=\dfrac{2}{3} x \sqrt{x}$

So the integration of root x is $\dfrac{2}{3} x \sqrt{x}$.