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How to Find the Cube Root of a Number? (With Formula & Examples)

how to find the cube root of a number

This post is also available in: हिन्दी (Hindi)

You know the volume of a rectilinear figure square = $\left(\text{side} \right) \times \left(\text{side} \right) \times \left(\text{side} \right)$, where $\text{side}$ is the length of edge(or side) of a square. If you want to find the length of an edge of a cube whose volume is known. How will you do that? In such cases, we find the cube root of a number. For example, if volume of a cube is $64 \text{cm}^3$, then the length of its edge(or side) is $4 \text{cm}$.

Let’s understand what is a cube root of a number, how to find the cube root of a number, and what are the properties of the cube root of a number.

What is Cube Root?

Whenever a number $a$ is multiplied three times, then the resultant number is known as the cube number of that number. The cube number for a number $a$ is represented as $a^{3}$ and is read as “$a$-cubed” (or) “$a$ to the power of $3$”. 

Therefore, the cube root of any number is another number that when multiplied by itself twice gives the number whose Cube Root is to be determined.

For example, in the case of a number $4$. We know that $4 \times 4 \times 4 = 64$. Hence, $64$ is called the cube of $4$. This means that $4$ is a cube root of $64$ and is written as $\sqrt[3]{64} = 4$.

In general, if $b$ is a cube root of $a$, then $\sqrt[3]{a} = b$. The radical sign $\sqrt[3]{}$ is used as a cube root symbol for any number with a small 3 written on the top left of the sign.

Another way to denote a cube root is to write $\frac{1}{3}$ as the exponent of a number., i.e., $\sqrt[3]{x} = x^{\frac{1}{3}}$.

Cube root is an inverse operation of the cube of a number.

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How to Find the Cube Root of a Number?

The cube root of any real number is obtained by either the prime factorization method or the estimation method when the number whose square root is to be found is a perfect cube number.

We can use any of these two methods to find the cube root of numbers.

  • Cube Root by Prime Factorization Method
  • Cube Root by Estimation Method

Cube Root by Prime Factorization Method

In this method, the number whose Cube Root is to be found is resolved completely into its prime factors. The identical prime factors are grouped such that three identical factors form one group. To determine the Cube Root, one factor from each group is collected and multiplied together. 

Step 1: The given number is completely resolved into its prime factors. It is always recommended to start the division with the lowest possible prime number and then go to the higher prime number when the quotient is not completely divisible by the number chosen. 

Consider a number $287496$ whose cube root is to be determined. The prime factorization of $287496$ is $2 \times 2 \times 2 \times 3 \times 3 \times 3 \times 11 \times 11 \times 11$. 

Step 2: Write the factors into powers of $3$.

$287496$ as the product of its primes is rewritten as:

$287496 = 2^{3} \times 3^{3} \times 11^{3}$

Step 3: The cube root of the number is found as the product of one factor taken from each group in step 2. So, the cube root of $287496$ is $2 \times 3 \times 11 = 66$.

Cube Root by Estimation Method

In the estimation method, we segregate the number into groups and then estimate its cube root. The steps to be followed while determining the cube root by estimation method are described below. 

Step 1: The given number is divided into groups of $3$ digits starting from the rightmost digit of the number. If any number is left out without forming a group of three, zeros are appended to its left to make it a group of $3$ digits. However, we must take care that the place value of the digit is not altered by appending zeros. 

Let us try finding out the cube root of $287496$.

So, to find the cube root of $287496$, we should divide the number into groups of three digits starting from the digit in the unit’s place. $287 \text{  } 496$

Step 2: From the first group starting from the right, note down the unit’s digit. The first rightmost group in step 1 is $496$ and the digit in its unit’s place is $6$.

Step 3: Estimate the digit in the unit’s place of the cube root of the given number using the table given below.

how to find the cube root of a number

The digit in the unit’s place obtained in step 1 is $6$ and hence the unit’s digit of the cube root of $287496$ is also $6$.

Step 4: Now, consider the second group from the right. Check the perfect cube numbers between which this number lies. Suppose the number in this group lies between $\text{A}^{3}$ and $\text{B}^{3}$ and is closer to $\text{B}^{3}$, then the ten’s digit of its cube root is considered as $B$, else it is $A$. 

In the given number $287496$, the second group of $3$ digits from the right is $287$.

This number lies between two perfect cube numbers $216$ and $343$ i.e. $6^{3}$ and $5^{3}$. Because $287$ is closer to $216$ i.e. $6^{3}$, the ten’s digit of the cube root of $287496$ is $6$.

The cube root of $287496$ found using the estimation method is $66$.

Difference Between Axiom, Postulate & Axiom

Practice Problems

  1. Find the cube root of the following numbers using the factorization method.
    • $74088$
    • $1157625$
    • $57066625$
    • $287496$
  2. Find the cube root of the following numbers using the estimation method.
    • $20346417$
    • $12326391$
    • $343000$
    • $52313624$

FAQs

What is a cube root of a number?

The cube root is the reverse of the cube of a number and is denoted by $\sqrt[3]{}$. For example, $\sqrt[3]{512}$, that is, the cube root of $512 = 8$ because when $8$ is multiplied thrice with itself, it gives $512$. In other words, since $8^{3} = 512$, we have $\sqrt[3]{512} = 8$.

What is the difference between a cube and a cube root?

When a number is multiplied by itself three times, then the product is the cube of the given number. 
For example, the cube of $4$ is $64$ because $4 \times 4 \times 4 = 64$.
The cube root of a number is a number whose cube is the given number. 
For example, the cube root of $64$ is $4$.

What is cube root used for?

Cube root is used to solve cubic equations. They are also used to find the side length of a cube if its volume is given.

Conclusion

The cube root of any number is another number that when multiplied by itself twice gives the number whose cube root is to be determined.  There are $2$ methods of finding the cube root of a number – Cube Root by Prime Factorization Method, and Cube Root by Estimation Method.

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