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 $A^3=x$, then the number $A$ is called a cube root of $x$. Symbolically, we can write it as

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

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:

  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 $1, \omega, \omega^2$, where $\omega=\dfrac{-1 \pm \sqrt{3}i}{2}$.
  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 $x^{1/3}$.

$\dfrac{d}{dx}(x^{1/3})$

$=\dfrac{1}{3} x^{1/3-1}$

$=\dfrac{1}{3} x^{-2/3}$

So the derivative of cube root x is 1/3 x^{-2/3}.

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Cube Root Integration

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

$\int x^{1/3} dx$

$=\dfrac{x^{1/3 +1}}{1/3+1}$

$=\dfrac{x^{4/3}}{4/3}$

$=\dfrac{3}{4} x^{4/3}$

$=\dfrac{3}{4} x \cdot x^{1/3}$

$=\dfrac{3}{4} x\sqrt[3]{x}$

So the integration of root x is $\dfrac{3}{4} x\sqrt[3]{x}$.

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