Cube Root of 2197

Dhristi JEE 2022-24

What is Cube Root?

If a number is multiplied three times, the resultant number is the cube of that number. On the other hand, the cube root of a number is the reverse process of the cube of a number and is denoted by ∛. In Mathematics, the cube root of a number (x) is the number (y) which satisfies the equation written below.


y³  = x


It can also be written in the form of


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


The symbol that is used to denote the cube root is \[\sqrt[3]{} \]


In this article, we will learn how to find the cube root of 2197, cube root of 2197 by estimation method, and cube root of 2197 by the prime-factorization method.


How to Find the Cube Root of 2197

The cube root of 2197 can be written as\[\sqrt[3] {2197}\]. It is a value  that can be obtained from the original number  by multiplying itself three times. As 2197 is a perfect cube, it will be very easy to calculate the cube root of 2197. Therefore, \[ \sqrt[3] {2197} \] =   \[\sqrt[3]{} \](13 * 13 * 13)   = 13. Hence, the cube root of 2197 is 13. Hence, the cube root is the inverse method of determining the cube of a number.


Cube root of 2197, i.e.,\[\sqrt[3] {2197}\] = 13


Let \[\sqrt[3] {2197}\] = p, then 2197 = p³. So, we will calculate the value of p. The  easiest method to find the cube root of  2197 is the estimation method although we will  study  the prime factorization method to calculate the cube root of 2197 for better understanding.

 

(Image will be uploaded soon)


Cube Root of 2197 by Estimation Method

To find the cube root of 2197 by estimation method, it is necessary for us to learn the cubes of natural numbers from 1 to 10.

Number

Cubes (n³)

1

1

2

8

3

27

4

64

5

125

6

216

7

343

8

512

9

729

10

1000


Let us now calculate the cube root of 2197 by using the estimation method in a step-by-step manner.

  1. Consider the digit given at the unit’s place of 2197.

The unit place 2197 has 7.

  1. Look at the cubes table given above. In the cube table, check the cube of which number has digit 7 at the unit’s place.

Clearly, we can see 3³ = 27.

It implies that the unit digit of the cube root of 2197 is 3.

  1. Now to calculate the tenth place of the cube root of 2197, we will overlook the last 3 digits of 2197, i.e., 197.

Considering 2 as a benchmark figure, we can see that the digit 2 falls between 1³ and 2³, i.e., between 1 and 8.

Here, we will consider the lowest numbers closest to 2 which is 1.

As 1³ = 1. Hence, the tenth place of the cube root of 2197 is 1.

Hence, the cube root of 2197 is 13.


Cube Root of 2197 by Prime Factorization Method

Here, we will discuss about the prime factorization method to determine the prime factors of 2197. After determining the prime factors of 2197, we will pair the factors in a group of three and write them as cubes. As the cubes of a number neutralise the cube root, we will get the desired value.


Now let us learn how to find the cube root of 2197 by the prime factorization method in a step-by-step manner.

  1. Calculate the prime factors of 2197.

If we try to divide 2197 by any prime number, we will find that the smallest prime factor which can divide 2197 is 13. Hence, we get

2197 = 13 * 13 * 13

  1. Now pair the factors in a group of 3 and express them in the form of cubes.

2197 = (13) = 13³ (As we know exponent law: ax × ay = ax + y)

  1. Now apply cube root to both left  and right sides  of the above  expression.

\[\sqrt[3] {2197}\] = \[ \sqrt[3]{3}(13^3) \]

The cube root gets neutralised by the cube of 13.

Hence, the cube root of 2197 is 13.


Solved Examples

1. Find the cube root of 4096 by the estimation method.

Solution:

  1. Let us consider the last 3 digits of 4096 as the first part and the remaining digit 4 as the second part.

  2. Now, look at the last digit of 4096, and with the help of the cubes table given above, find the cube of a digit (from 0 to 9) that has the last digit 6.

6³ = 216, the last digit of 216 is 6.

So, the unit place of the cube root of 4096 is 6.

  1. Now, look at the first digit of the second part of the given number, i.e., 4.

The number 4 lies in between the cubes of 1 and 2 (i.e., in between  1 and 8). Consider the lowest number among the two given numbers (1 and 2). The lowest number among the two given numbers is 1.

Hence, the tenth digit of the cube root of 4096 is 1.

Therefore, the cube root of 4096 is 16.


2. Determine the cube root of 343 through the prime factorization method.

Solution:

  1. First, we will find the prime factors of 343.

So, the prime factors of 343 are  7 × 7 × 7. 

  1. Now, we will pair similar factors in a group and represent them as cubes.

           343 =  (7 * 7 * 7) 

           343 = 73

  1. Now, we will apply the cube root on both the left and right sides of the above expression.

\[\sqrt[3] {343}\] = \[ \sqrt[3]{3}(7^3) \]

Here, the cube root gets neutralised by the cube of 7.

Hence, the cube root of 343 is 7.


Quiz Time

  1. What will be the cube root of .000216?

  1. 6

  2. .06

  3. 77

  4. 87

Ans: b


  1. Find the number which is not a perfect cube among the numbers given below.

  1. 1331

  2. 216

  3. 243

  4. 512

Ans: c


  1. What will be the cube root of 1.331?

  1. 0.11

  2. 0.011

  3. 11

  4. 1.1

Ans: d

FAQs on Cube Root of 2197

1. What is a perfect cube?

A perfect cube is a number that can be written as the product of three similar integers. In other words, a perfect cube is a number that can be formed by cubing an integer. Each perfect cube is considered itself as an integer.


Also, the perfect cube has the same symbol as the number you start with.

  • The cube of a positive integer is always a positive integer.

  • The cube of a negative integer is always a  negative integer.

  • The cube of zero is zero.

For example, 8 is a perfect cube because 8 can be written as 


23 = 2 × 2 × 2.


Also, 64 is a perfect cube as 64 can be expressed as


43 = 4 × 4 × 4 = 64.

2. How to find the cube root of a number?

The cube root is an inverse process of determining the cube of a number. The symbol used to represent the cube root of a number is ∛. Let us learn with an example given below.


To calculate the cube root of 2197, we will find a number which when multiplied thrice with itself obtains 2197. We can write it as,

 

2197 = 13 ×13 ×13


Taking cube root on both the sides:


or \[\sqrt[3] {2197}\] = \[ \sqrt[3]{3}(13^3) \]


Hence, the cube root of 2197 is 13.


Please note that we will consider only the positive values cube roots of any given natural number.

Comment