Cube and Cube Root: Definition, Formula, How to Find, Examples

The cube root of a number is an important concept like square roots in the number system. Note that the cube root is the inverse method of finding cubes. In this section, we will discuss about the cube root of a number.

What is cube and cube root? The number obtained by multiplying a given number two times by itself is called the cube of that given number. So m×m×m=m³ is the cube of m. Here the number m is called the cube root of m³.

Definition of Cube of a Number

A number is called the cube of a number m if the number is a product of three numbers of m‘s. Thus m³ is the cube of m.

For example, 8 is the cube of 2 as 8=2×2×2.

Similarly, 64 is the cube of 4 as 64=4³.

Definition of Cube Root

A number x is called a cube root of a number y if the following relation is satisfied:

\[x \times x \times x=y^3.\]

Thus, if y=x³ then we say x is the cube root of y.

Note that 8=2³, so we can say that 2 is a cube root of 8.

Cube Root Symbol

Symbol of cube root: the cube root is represented by the symbol “$\sqrt[3]{}$”.

As the cube root of 8 is 2, mathematically we write it as follows:

$\sqrt[3]{8}=2$

Cube Root Examples

More examples of cube roots:

• As 3³ = 3×3×3 = 27, we have $\sqrt[3]{27}$=3 by the definition of cube roots.

• We have 4³ = 4×4×4 = 64. Thus $\sqrt[3]{64}$=4, so the cube root of 64 is 4.

• Note that 5³ = 5×5×5 = 125. Therefore $\sqrt[3]{125}$=5.

Rules of Cube Root

The following are the properties of cube roots.

• $x^{1/3}=\sqrt[3]{x}$
• $\sqrt{x^3}=x$
• $a\sqrt[3]{x}+b\sqrt[3]{x}=(a+b)\sqrt[3]{x}$
• $a\sqrt[3]{x}-b\sqrt[3]{x}=(a-b)\sqrt[3]{x}$
• $\sqrt[3]{x \times y}=\sqrt[3]{x} \times \sqrt[3]{y}$
• $\sqrt[3]{\frac{x}{y}}=\frac{\sqrt[3]{x}}{\sqrt[3]{y}}$   if $y \neq 0$

Cube Root Formula

The cube root $\sqrt[3]{y}$ can written as $\sqrt[3]{y}=y^{1/3}$ by the rule of indices. If the number $y$ is a perfect cube, that is, $y=x \times x \times x$ for some number $x$ then  we use the formula below to find the cube root of $y:$

\[\sqrt[3]{y}=\sqrt[3]{x \times x \times x}=x.\]

Cube Root Table

Perfect cube table:

Also Read: Fourth root of 81

Simplifying Cube Roots 

Let us now learn how to simplify cube roots. We will simplify cube root of 24.

Question: Simplify cube root of 24.

Solution: At first, we will factorize 24. See that

24 = 2×2×2×3

Taking cube root on both sides, we get

$\sqrt[3]{24}=\sqrt[3]{2 \times 2 \times 2 \times 3}$

$=\sqrt[3]{2 \times 2 \times 2} \times \sqrt[3]{3}$  $[\because \sqrt[3]{x \times y}=\sqrt[3]{x} \times \sqrt[3]{y}]$

$=2 \times \sqrt[3]{3}$  $[\because \sqrt[3]{a \times a \times a}=a]$

$=2\sqrt[3]{3}$

So the square root of 24 is $2\sqrt[3]{3}$ (YouTube video 0f cube root of 24).

Methods of Finding Cube Root

How to find the cube root of a number?

Addition or Subtraction of Cube Roots 

How to add/subtract two square roots? The following steps have to be followed.

Step 1: First, we express the cube roots into their simplified forms as above like cube root of 24.

Step 2: If the simplified forms contain only the cube roots $\sqrt{x}$, then apply the formula $a\sqrt[3]{x}\pm b\sqrt[3]{x}=(a\pm b)\sqrt[3]{x}$ to get the desired sum or difference. If they have different radicals then we keep them as it is.

For example, add $\sqrt[3]{24}$ and $\sqrt[3]{81}$

Question: Find $\sqrt[3]{24}+\sqrt[3]{81}$

Solution: Firstly, we will simplify both cube roots.

From above $\sqrt[3]{24}=2\sqrt[3]{3}$

Also, $\sqrt[3]{81}=\sqrt[3]{3 \times 3 \times 3 \times 3}$ $=3\sqrt[3]{3}$

So $\sqrt[3]{24}+\sqrt[3]{81}$ $=2\sqrt[3]{3}+3\sqrt[3]{3}$ $=5\sqrt[3]{3}.$

Similarly, $\sqrt[3]{50}-\sqrt[3]{81}$ $=2\sqrt[3]{3}-3\sqrt[3]{3}$ $=-\sqrt[3]{3}$

Multiplication or Division of Cube Roots 

How to multiply or divide two cube roots? We will follow the below steps.

Step 1: We write the cube roots into their simplified forms.

Step 2: Then applying the formula $a\sqrt[3]{x} \times b\sqrt[3]{y}=ab\sqrt[3]{xy}$ we get the desired multiplication. To divide the cube roots, we will use the formula $a\sqrt[3]{x} \div b\sqrt[3]{y}=a/b \frac{\sqrt[3]{x}}{\sqrt[3]{y}}.$

Cube Root of a Decimal

How to find the cube root of decimals? We have to follow the below steps to compute the cube root of a decimal number.

Step 1: At first, we need to express the decimal number as a fraction.

Step 2: Then we use the formula $\sqrt[3]{\frac{x}{y}}=\frac{\sqrt[3]{x}}{\sqrt[3]{y}}.$

Step 3: Now we will simplify the cube roots of $x$ and $y$ and then we put their values in the fraction obtained in step 2.

Step 4: Simplifying the fraction obtained in step 3, we will get the cube root of the given decimal number.

To understand the above method, we provide an example here.

Question: Find $\sqrt{0.008}$

Solution:

Note that $0.008=\frac{8}{1000}$

Taking square root we get

$\sqrt[3]{0.008}=\sqrt[3]{\frac{8}{1000}}$ $=\frac{\sqrt[3]{8}}{\sqrt[3]{1000}}$ $=\frac{2}{10}$ $=0.2$

So the square root of 0.008 is 0.2.

Cube Root of a Complex Number

The computation of the cube of a complex number is not as simple as the method of finding the cube root of a real number. We know that the general form of a complex number is $a+ib,$ where both $a$ and $b$ are real numbers.

Cube Root as a Function

Note that the cube root of a number takes three values. So it will not make a function. But if we define $\sqrt{x^3}=+x,$ then it will definitely make a function.  Now define a function

\[f: \mathbb{R} \to \mathbb{R} \,\, \text{by} \,\, f(x)=\sqrt[3]{x}.\]

The above function $f$ is one-to-one but not onto.

Generalization of Cube Roots

It is known that the cube root $x$ of the number $y$ satisfies the relation $y=x^3.$ Symbolically, we write $\sqrt[3]{y}=x.$ This can be generalized in a broad manner.

Fourth root: If $x^4=y$ then $x$ is called a fourth root of $y$ and we write it as $\sqrt[4]{y}=x.$ In exponent form $\sqrt[4]{y}=y^{1/4}.$

Fifth root: If $x^5=y$ then $x$ is called a fifth root of $y.$ It is written as $\sqrt[5]{y}=x.$ In exponent form $\sqrt[5]{y}=y^{1/5}.$

n-th root: If $x^n=y$ then $x$ is called a n-th root of $y.$ Note that $\sqrt[n]{y}=x.$ In exponent form $\sqrt[n]{y}=y^{1/n}.$

Polynomial root: Let $f(x)$ be a polynomial with a root $c$. Then we must have $f(c)=0.$ This type of roots are classified as polynomial roots.

Applications of Cube Root 

The computation of cube roots has many applications in several branches of mathematics; such as

• Equation solving (for example, solve x³=8)
• Polynomial
• Numerical analysis
• Geometry (for example, to find the side of a cube if the volume is known)

Solved Problems of Cube Roots

Question: Find the cube root of $125.$

Solution:

Note that $125=5 \times 5 \times 5$

So $\sqrt[3]{125}=\sqrt{5 \times 5 \times 5}$

$=5$  $[\because \sqrt[3]{x \times x \times x}=x]$

So the value of the cube root of 125 is 5.

Cube Root of Numbers

Cube root of 8	Cube root of 16
Cube root of 24	Cube root of 27
Cube root of 32	Cube root of 40
Cube root of 48	Cube root of 54
Cube root of 56	Cube root of 64
Cube root of 72	Cube root of 80
Cube root of 81	Cube root of 88
Cube root of 96	Cube root of 108
Cube root of 120	Cube root of 125
Cube root of 135	Cube root of 216
Cube root of 343	Cube root of 512
Cube root of 721	Cube root of 1000