Cube Root of x: Definition, Symbol, Graph, Properties, Derivative, Integration

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 of x

Cube Root Definition

If A3 =x, then the number A is called a cube root of x. Symbolically, we can write it as

A=Cube root x

Cube root Symbol: 

Note: The cube root of x can be written as x1/3, that is,  Cube root x= x1/3.

Cube Root Example

As 23 =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 33=27 , the number 3 is a cube root of 27.
  • As 43 =64, the number 4 is a cube root of 64.
  • As 53 =125, the number 5 is a cube root of 125.

Cube Root Graph

Let $y=$Cube root x
Taking cubes on both sides, we get that
y3 =x.
The graph of cube root is plotted below:
Source: Wikipedia

Cube Root Properties

Below are the properties of cube roots:
  1. 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,
  2. If $a, b (\neq 0)$ be two real numbers. Then we have
  3. 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.
  4. The cube root of 0 is 0.
  5. The cube roots of 1 are 1, ω, ω2, where ω=$\dfrac{-1 \pm \sqrt{3}i}{2}$.
  6. 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 Cube root x will be equal to
$=\dfrac{1}{3} x^{1/3-1}$
$=\dfrac{1}{3} x^{-2/3}$
So the derivative of cube root x is 1/3 x^{-2/3}.
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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 Cube root x$=x^{1/3}$, by the above power rule of integration, the integration of cube root x is
$\int x^{1/3} dx$
$=\dfrac{x^{1/3 +1}}{1/3+1}$
$=\dfrac{3}{4} x^{4/3}$
$=\dfrac{3}{4} x \cdot x^{1/3}$
$=\dfrac{3}{4} x\sqrt[3]{x}$
So the integration of root x is $\dfrac{3}{4} x\sqrt[3]{x}$.
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