How to Use the Calculator?
To use the calculator, simply enter a value in either the Number field or the Cube Root field, and the other field will update automatically as you type.
Use the Number field when you want to find the cube root of a number, or use the Cube Root field when you want to find the number from its cube root.
What Is a Cube Root?
A cube root is a number that is multiplied by itself three times to give another number.
The idea of cube root is closely connected to the idea of a cube. If a cube has side length $a$, then its volume is found by multiplying $a$ by itself three times.
$$V=a^3$$
This means that if we know the side length of a cube, we can find its volume by cubing the side length.
But sometimes we are given the volume of the cube and asked to find the side length. In that case, we use the cube root. So, cube root is the reverse operation of cubing a number.
For example, if a cube has side length $2$ units,
its volume is:
$$2^3=2\times 2\times 2=8$$
So, the cube root of $8$ is $2$.
$$\sqrt[3]{8}=2$$
The symbol $\sqrt{\ }$ is called the Radical Sign. The number inside the radical sign is called the Radicand. The small $3$ written above the radical sign shows that we are finding the cube root, not the square root.
Perfect Cubes
A perfect cube is a number that can be written as the cube of an integer. In other words, if a number is obtained by multiplying an integer by itself three times, it is called a perfect cube.
For example:
$$1^3=1\times 1\times 1=1$$
$$2^3=2\times 2\times 2=8$$
$$3^3=3\times 3\times 3=27$$
$$4^3=4\times 4\times 4=64$$
$$5^3=5\times 5\times 5=125$$
So, $1$, $8$, $27$, $64$, and $125$ are perfect cubes. These numbers have exact cube roots. That means their cube roots are whole numbers.
Cube Root of Negative Numbers
One special feature of cube roots is that negative numbers can also have real cube roots. This is different from square roots, where the square root of a negative number is not a real number.
For cube roots, the rule is simple: the cube root of a negative number is negative.
For example:
$$(-2)^3=(-2)\times(-2)\times(-2)=-8$$
Therefore:
$$\sqrt[3]{-8}=-2$$
Let us understand why. When two negative numbers are multiplied, the result is positive. But when a third negative number is multiplied, the result becomes negative again.
$$(-2)\times(-2)=4$$
$$4\times(-2)=-8$$
So, $(-2)^3=-8$. That means the cube root of $-8$ is $-2$.
This is an important rule to remember: positive numbers have positive cube roots, and negative numbers have negative cube roots.
Cube Root of Fractions
To find the cube root of a fraction, we take the cube root of the numerator and the cube root of the denominator separately.
$$\sqrt[3]{\frac{a}{b}}=\frac{\sqrt[3]{a}}{\sqrt[3]{b}}$$
For example, find:
$$\sqrt[3]{\frac{8}{27}}$$
We know:
$$\sqrt[3]{8}=2$$
and:
$$\sqrt[3]{27}=3$$
Therefore:
$$\sqrt[3]{\frac{8}{27}}=\frac{2}{3}$$
Cube Root of Decimal Numbers
Cube roots of decimals can often be solved by writing the decimal as a fraction. This makes the problem easier to understand.
For example, find:
$$\sqrt[3]{0.008}$$
First, write $0.008$ as a fraction.
$$0.008=\frac{8}{1000}$$
Now take the cube root.
$$\sqrt[3]{0.008}=\sqrt[3]{\frac{8}{1000}}$$
Since:
$$\sqrt[3]{8}=2$$
and:
$$\sqrt[3]{1000}=10$$
we get:
$$\sqrt[3]{0.008}=\frac{2}{10}=0.2$$
So:
$$\sqrt[3]{0.008}=0.2$$
Methods to Find the Cube Root of a Number
There are several useful methods for finding the cube root of a number. Three important methods are:
- Repeated subtraction method
- Estimation method
- Prime factorisation method
Repeated Subtraction Method
The repeated subtraction method is a simple way to find the cube root of a perfect cube. In this method, we start with the number whose cube root we want to find. Then we subtract special numbers in order from the given number until we reach $0$. These special numbers rely on a specific mathematical pattern: $3n(n-1) + 1$. The number of subtractions tells us the cube root.
Let us find $\sqrt[3]{64}$ using repeated subtraction. We start with $64$ and subtract the cube-difference numbers in order.
$$64-1=63$$
$$63-7=56$$
$$56-19=37$$
$$37-37=0$$
We made $4$ subtractions before reaching $0$. Therefore, the cube root of $64$ is $4$.
$$\sqrt[3]{64}=4$$
This method is easy to understand because it shows how cube numbers are built step by step. However, it can take more time for larger numbers. For example, finding $\sqrt[3]{1000}$ by this method would require $10$ subtractions. So, this method is best for small perfect cubes or for understanding the pattern behind cube numbers.
Estimation Method
The estimation method is used when a number is not a perfect cube. It helps us find a reasonable answer and understand the size of the cube root without depending only on a calculator.
For example, let us estimate:
$$\sqrt[3]{50}$$
First, we find the perfect cubes closest to $50$. We know:
$$3^3=27$$
$$4^3=64$$
Since $50$ is between $27$ and $64$, its cube root must be between $3$ and $4$.
$$3<\sqrt[3]{50}<4$$
Now we decide whether $\sqrt[3]{50}$ is closer to $3$ or $4$. Since $50$ is closer to $64$ than to $27$, the cube root of $50$ is closer to $4$ than to $3$. A good first estimate might be $3.7$.
Let us check $3.7$ by cubing it.
$$3.7^3=3.7\times 3.7\times 3.7=50.653$$
Since $50.653$ is slightly greater than $50$, $3.7$ is a little too high. Now let us try $3.6$.
$$3.6^3=3.6\times 3.6\times 3.6=46.656$$
Since $46.656$ is less than $50$, $3.6$ is too low. This tells us that $\sqrt[3]{50}$ lies between $3.6$ and $3.7$.
$$3.6<\sqrt[3]{50}<3.7$$
If we want a closer estimate, we can try $3.68$.
$$3.68^3=49.836032$$
This is slightly less than $50$, so $3.68$ is a little low. Now try $3.69$.
$$3.69^3=50.243109$$
This is slightly greater than $50$, so $3.69$ is a little high. Therefore, $\sqrt[3]{50}$ lies between $3.68$ and $3.69$.
$$3.68<\sqrt[3]{50}<3.69$$
A good estimate is:
$$\sqrt[3]{50}\approx 3.684$$
Prime Factorisation Method
Prime factorization is one of the best methods for finding the cube root of a large perfect cube. In this method, we write the number as a product of prime factors. Then we make groups of three equal factors. From each group of three equal factors, we take one factor outside the cube root.
Let us find:
$$\sqrt[3]{216}$$
First, write $216$ as a product of prime factors.
$$216=2\times 2\times 2\times 3\times 3\times 3$$
Now group the same factors in threes.
$$216=(2\times 2\times 2)\times(3\times 3\times 3)$$
Each group of three equal factors gives one factor outside the cube root.
$$\sqrt[3]{216}=2\times 3=6$$
So:
$$\sqrt[3]{216}=6$$
Choosing the Right Method
Each method for calculating a cube root serves a specific purpose.
To find the cube root of a number, we first determine whether the number is a perfect cube. If it is a perfect cube, methods such as repeated subtraction and prime factorisation allow us to obtain an exact, integer answer. If the number is not a perfect cube, the estimation method proves to be highly useful.
All three of these methods yield the same result; what matters is how correctly you apply them.
