Cube Root Formula: Definition, Explanation & Examples

Cube root is the inverse operation of cubing a number. The cube of any number is the product obtained when the number is multiplied by itself three times. The cube root formula is used to give the cube root of any number. The correct application of cube root makes it easier for students to calculate difficult sums, consuming less amount of time. This article will help students understand the technique to be used to calculate cube root and help them in solving the problems based on the same.

Learn Cubes From 1 To 30

Cube Root: Baisc Details

For learning and understanding the cube root formula, it is mandatory to know about the cube formula.
All cube numbers can be written as \(a^{3}\), and
\(a^{3}=a \times a \times a\)

Have you seen a Rubik’s cube? Look at the picture of Rubrik’s cube given below.

It has \(3 \times 3 \times 3\), i.e., \(27\) cubes.
Thus, \(27\) is a cube number.

The cube of an odd number is odd, and the cube of an even number is even. If a number has the digit \(3\) in the unit’s place, the cube of that number will have the digit \(7\) in the units place, and if a number has the digit \(7\) in the unit’s place, the cube of that number will have the digit \(3\) in the units place. Remember that the square of any number, positive or negative, is always positive. The cube of a positive number is always positive, but a cube of a negative number is always negative.

Cube Root: Defination

Suppose the volume of a cube is \(125 \mathrm{~cm}^{3}\). What would be the length of its sides? To get the length of the side of the cube, we need to know a number whose cube is \(125\).

We are aware that the cube root formula is the inverse of the cube formula. In the cube formula, we multiply a number three times to get its cube, so to find the cube root of a number, we break down the number to be expressed as a product of three equal numbers to get the cube root. Cube root is the number that needs to be multiplied three times to get the original number.

Cube Root: Symbol

The symbol for cube root is \(\sqrt[3]{ }\).
For example, \(2^{3}=8\), read as two cubed is equal to eight.
Then, \(\sqrt[3]{8}=2\), read as the cube root of \(8\) is equal to \(2\).
Again, \(10^{3}=1000\), read as ten cubed is equal to one thousand.
Then, \(\sqrt[3]{1000}=10\), read as the cube root of \(1000\) is equal to \(10\).
Thus, we can define a perfect cube as a number whose cube root is an integer.

Finding the Cube Root-Inverse Operation of Finding Cube

Numbers whose cube roots are integers are called perfect cube numbers. For example, \(8,27,64,125,216\) are perfect cube numbers because their cube roots are \(2,3,4,5,6\) respectively. If the number \(x\) is not a perfect cube, \(\sqrt[3]{x}\) is called a radical, \(x\) is called a radical, and \(3\) is called the radical index.

We can find the cube of a number by the prime factorisation method. 

The following are the rules to be followed.

1. Find the prime factors of the given number whose cube root is to be determined.
2. Express the given number as the product of prime factors.
3. Make groups of \(3\) equal prime numbers.
4. Take one prime number from each group of three equal prime numbers and find their product.
5. This product will be the required cube root of the given number.

Let us understand using some examples.

Example 1: Find the cube root of \(2744\).

Solution: Let us write \(2744\) as the product of its primes factors.
\(2744=2 \times 2 \times 2 \times 7 \times 7 \times 7\)
Each factor appears in a group of three. So, we can write,
\(2744=2^{3} \times 7^{3}=14^{3}\)
Therefore, \(\sqrt[3]{2744}=\sqrt[3]{14^{3}}=14\)

Example 2: Find the cube root of \(42875\).

Solution: Let us write \(42875\) as the product of its primes factors.
\(42875=5 \times 5 \times 5 \times 7 \times 7 \times 7\)
Each factor appears in a group of three. So, we can write,
\(42875=5^{3} \times 7^{3}=35^{3}\)
Therefore, \(\sqrt[3]{42875}=\sqrt[3]{35^{3}}=35\).
The cube root of the number from \(1\) to \(20\) is shown below in the table.

Cube Root of a Negative Perfect Cube

Let \(x\) be the positive integer. Then, \((-x)\) is the negative integer and
\((-x)^{3}=-x^{3}\)
Therefore,
\(\sqrt[3]{-x^{3}}=-x\)
Thus, the cube root of \((-x)^{3}=-\left(\right.\) cube root of \(x^{3}\))
Hence, \(\sqrt[3]{-x}=-\sqrt[3]{x}\)

Cube Root of a Number: Technique to Calculate

We can use estimation to find the cube root of a cube number. In mathematics, estimation is not guesswork. Instead, it involves the use of logical thinking to refine an answer. 

The following are the rules to be followed, they are;

1. Make groups of three digits, beginning with an extreme right digit of the given number. For example, \(15625\) can be grouped as \(\underline{15} \,\underline{625}\).
   The first group will be \(\underline{625}\) and the second group will be \(\underline{15}\).
2. See the digit at the unit’s place in the first group of three digits in the given number. Then, identify the digit at the unit’s place in the given number’s cube root using the following table.

So, the unit place of \(15625\) will be \(5\) as the unit place in the first group of three digits of \(15625\) is \(5\).

1. Strike out the digits at units, tens, and hundreds places, i.e., the three digits from the right. If no digits are left out, then the digit obtained in step \(2\) is the required cube root. If digits are left out, proceed further.
2. Consider the number left after striking out the three digits in step \(2\). Identify the largest one-digit number whose cube is less than or equal to this left-out number. This digit will be the tens digit of the required cube root. The number left out in \(15625\) is \(\underline {15} \).
   The largest one-digit number whose cube is less than or equal to \(12\) is \(2^{3}=8\).
3. Now write down the number whose units digit is the number obtained in step \(1\), and the tens digit is the number obtained in step \(3\), then write the required cube root.
   So, \(\sqrt[3]{15625}=25\)

Let us understand this method using another example.

Example: Find \(\sqrt[3]{17576}\).

Solution: The first step is to group the digits in three, starting from the units place of the given number.
For the number \(17576\), grouping the digits will make it \(\underline{17}\, \underline{576}\).
The digits on the right will be referred to as the first group, i.e., \(576\). Thus, the second group is \(17\).
The first group gives us the units digit of the required cube root, and we will find it by estimation. Since the unit digit in the first group is 6, the required number must be \(6\).
Now, let us have a look at the second group, i.e., \(17\). Thus number will give us a pointer towards the tens digit of the required number.
We know that, \(2^{3}=8\) and \(3^{3}=27\). Thus, \(2^{3}<17<3^{3}\).
This tells us that the required cube root lies between \(20\) and \(30\). We already figured out that the unit digit is \(6\)
Therefore, \(\sqrt[3]{17576}=26\)

Cube Root Calculator

A cube root calculator is a formula used to find the cube root of positive and negative integers. Given a number \(a\), the cube root of \(a\) is a number \(x\) such that \(x^{3}=a\).

Cube root formula trick

Let us understand with the help of an example.
Find the cube root of \(778688\).
Now, first group the digit in threes, starting from the unit’s place.
As we can see, the unit digit in the first group is \(8\). Thus the unit’s digit of the required number is \(2\).
Now, take the second group, i.e., \(778\).
We know that \(9^{3}=729\) and \(10^{3}=1000\). Thus, \(9^{3}<778<10^{3}\).
This tells us the cube root lies between \(90\) and \(100\). We already know that the unit digit is \(2\).
Therefore, the cube root will be \(92\).

Solved Examples – Cube Root Formula

Q.1. The volume of a cube is \(3375 \mathrm{~cm}^{3}\). Find the length of a side of the cube by prime factorisation.
Ans: Volume of a cube \( = {\text{length}} \times {\text{breadth}} \times {\text{height}}\)
Now, length of a cube \(=\sqrt[3]{\text { Volume }}=\sqrt[3]{3375}\)
\(2275=3^{3} \times 5^{3}=15^{3}\)
Thus, \(\sqrt[3]{3375}=15\)
Hence, the length of a side of the cube is \(15 \mathrm{~cm}\).

Q.2. Find the cube root of \((-1000)\).
Ans: We know that \(\sqrt[3]{-1000}=-\sqrt[3]{1000}\)
Resolving \(1000\) into prime factors, we get,
\(1000=2 \times 2 \times 2 \times 5 \times 5 \times 5\)
\(=(2 \times 2 \times 2) \times(5 \times 5 \times 5)\)
Therefore, \(\sqrt[3]{1000}=2 \times 5=10\)
Thus, \(\sqrt[3]{-1000}=-\sqrt[3]{1000}=-10\)
Hence, the cube root of \((-1000)=-10\).

Q.3. Evaluate \(\sqrt[3]{125 \times 64}\)
Ans: We have, \(\sqrt[3]{125 \times 64}=\sqrt[3]{125} \times \sqrt[3]{64}\)
\(=\sqrt[3]{5 \times 5 \times 5} \times \sqrt[3]{4 \times 4 \times 4}\)
\(=5 \times 4=20\)

Q.5. Find the cube root of \(\sqrt[3]{\frac{216}{2197}}\)
Ans: \(\sqrt[3]{\frac{216}{2197}}=\frac{\sqrt[3]{216}}{\sqrt[3]{2197}}\)
\(=\frac{\sqrt[3]{6 \times 6 \times 6}}{\sqrt[3]{13 \times 13 \times 13}}=\frac{6}{13}\)
Hence, the required answer is \(\frac{6}{13}\).

Q.5. Oranges have been put in a box in the form of a cube. If the number of oranges is \(216\), then how many layers of oranges are in the box.
Ans: Let the number of layers of oranges in the box \(=a\).
Since oranges have been put in the form of a cube, therefore the number of rows in each column \(=a\).
There number of oranges in each layer \(=a \times a=a^{2}\)
Thus, the total number of apples in the box \(=a^{2} \times a=a^{3}\)
According to the given question, \(a^{3}=216\)
\(=\sqrt[3]{216}=\sqrt[3]{6 \times 6 \times 6}=6\)
Hence, the number of layers in the box \(=6\)

Summary

In this article, we made ourselves aware of cube, cube root and then learned the method of factorisation to find the cube root of a number. We also learned to find the cube root by estimation and looked at the cube root formula.

Frequently Asked Question – Cube Root Formula

Frequently asked questions related to cube root formula are listed as follows:

Q.1. What is a cube root?
Ans: In the cube formula, we multiply a number three times to get its cube, so to find the cube root of a number, break down the number to be expressed as a product of three equal numbers and thus, we get the cube root. Cube root is the number that needs to be multiplied three times to get the original number. 

Q.2. What is the difference between cube root and square root?
Ans: Cube root is the number that needs to be multiplied three times to get the original number, whereas the square root of a given number is that number whose square is the given number.

Q.3. What is the cube root of \(69\)?
Ans: The cube root of \(69=\sqrt[3]{69}=4.101\).

Q.4. What is the cube root of \(64\)?
Ans: The cube root of \(\sqrt[3]{64}=\sqrt[3]{4 \times 4 \times 4}=4\)

Q.5. Can we find the cube root of negative numbers?
Ans: Yes, we can find the cube root of a negative number.
The cube root of \({\left({ – x} \right)^3} =  – \) (cube root of \({x^3}\))
Hence, \(\sqrt[3]{-x}=-\sqrt[3]{x}\)

Q.6. What is the formula of cube and cube root?
Ans: Cube of a number \(x=x \times x \times x\)
For example, let us take the number \(43\). We know that, \(4 \times 4 \times 4=64\). Hence, \(64\) is called the cube of \(4\).
The cube root of a number is the reverse process of the cube of a number. If \(m\) is the cube of \(n\), then \(n=\sqrt[3]{m}\).
For example: \(\sqrt[3]{125}=5\).

learn Cube Roots of Unity and Their Properties

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