The cube root of x is an algebraic function. In this article, we will learn its definition, graph, various properties, etc. We will also learn how to find its derivative and integration.
Cube Root Definition
If $A^3=x$, then the number $A$ is called a cube root of $x$. Symbolically, we can write it as
A=
Cube root Symbol: 
Note: The cube root of $x$ can be written as $x^{1/3}$, that is, $=x^{1/3}$.
Cube Root Example
As $2^3=8$, by the above definition, we can say that $2$ is a cube root of $8$. More examples of cube roots are given below:
- As $3^3=27$, the number $3$ is a cube root of $27$.
- As $4^3=64$, the number $4$ is a cube root of $64$.
- As $5^3=125$, the number $5$ is a cube root of $125$.
Cube Root Graph
Let $y=$
Taking cubes on both sides, we get that
$y^3=x$.
The graph of cube root is plotted below:
Source: Wikipedia
Cube Root Properties
Below are the properties of cube roots:
- The product of the cube roots of two numbers is the same as the cube root of the product of those two numbers. That is,
- If $a, b (\neq 0)$ be two real numbers. Then we have
- Cube root of a perfect cube is always an integer. Perfect cube is such a number that is the cube of some integer. For example, the cube root of $8$ is $2$, so $8$ is a perfect cube number.
- The cube root of $0$ is $0$.
- The cube roots of $1$ are $1, \omega, \omega^2$, where $\omega=\dfrac{-1 \pm \sqrt{3}i}{2}$.
- The cube root function is an odd function.
Cube Root Derivative
To find the derivative of cube root x, we will use the power rule: $\dfrac{d}{dx}(x^n)=nx^{n-1}$. We know that cube root of $x$ can be written as $x^{1/3}$.
Thus, by the above power rule, the derivative of will be equal to
will be equal to$\dfrac{d}{dx}(x^{1/3})$
$=\dfrac{1}{3} x^{1/3-1}$
$=\dfrac{1}{3} x^{-2/3}$
So the derivative of cube root x is 1/3 x^{-2/3}.
Cube Root Integration
To find the integration of root x, we will use the power rule of integration: $\int x^n dx=\dfrac{x^{n+1}}{n+1}$.
As $=x^{1/3}$, by the above power rule of integration, the integration of cube root x is
$=x^{1/3}$, by the above power rule of integration, the integration of cube root x is$\int x^{1/3} dx$
$=\dfrac{x^{1/3 +1}}{1/3+1}$
$=\dfrac{x^{4/3}}{4/3}$
$=\dfrac{3}{4} x^{4/3}$
$=\dfrac{3}{4} x \cdot x^{1/3}$
$=\dfrac{3}{4} x\sqrt[3]{x}$
So the integration of root x is $\dfrac{3}{4} x\sqrt[3]{x}$.