Cube Root of 2197

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How to Find the Cube Root of 2197?

The cube root of 2197 is written as ∛2197 . It is a value which obtains the original number on getting multiplied by itself three times. Generally, the cube root of a number denotes the root of a cubed number. In other words, if x³ = y then ∛y = x. Hence, the cube root is the inverse method of determining the cube of a number.

Cube root of 2197 , ∛2197= 13


Let ∛2197 = p, then 2197 = p³. Here, we will calculate the value of p. The fastest and easiest method to find the cube root of a four- digit number 2197 is the estimation method. Although, we will also study a prime factorization method to find the cube root of four -digit number 2197 for better understanding. In this article, we will study, how to find the cube root of 2197, cube root of 2196 by examination method, cube root of 2196 by prime - factorization method

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Cube Root Definition

The cube root of any number says ‘x’ is the number say ‘y’ which satisfies the equation written below.

y³  = x

It can be written in the form of

x = ∛y

The symbol used to denote the cube root is ∛ For example: ∛2197 = ∛(13 * 13 * 13) = 13. As 2197 is a perfect cube, it is easy to determine the cube root of 2197. Hence, the cube root of 2197 is 13.


What are the Different  Methods to Find the Cube Root of 2197?

There are two methods to find the cube root of 2197. These two methods are

  1. Estimation Method

  2. Prime Factorization Method

We will study the estimation Method and prime factorization method in detail through this article.


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 estimation method step by step.

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

The unit place 2197 is 7.

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

Clearly, we can see 3³ = 27

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

∛2197 = 3

So, we have got the unit digit 2197 and i.e. 3

3. 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 nearest to 2 which is 1.

As 1³ = 1. Hence, 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 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 to write them as cubes. As the cubes of a number neutralize the cube root, therefore we will get the desired value.

Let us learn  to find the cube root of 2197 by prime factorization method step by step

1. Calculate the prime factors of 2197

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

2197 = 13 * 13 * 13

2. Pair the factors in a group of three and express them in the form of cubes

2197 = (13) = 13³ [ Through exponent law: ax × ay = ax + y]

3. Apply cube to both left side and right side for the above given expression.

∛2197 = ∛(13)³

The cube root gets neutralized by the cube of 13

Hence, the cube root of 2197 is 13


Solved Examples

  1. Find the cube root of 4096 by 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 and i.e. 4.

The number 4 lies in between the cubes of 1 and 2 ( i.e. in between the 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.

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

Solution:

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

Prime Factors of 343 is  7×7 × 7 

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

          343 =  (7 * 7 * 7) 

          343 = 7

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

∛343 = ∛7³ 

Here, the cube root gets neutralized 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

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

  1. 1331

  2. 216

  3. 243

  4. 512

3. What will be the cube root of 1.331?

  1. 0.11

  2. 0.011

  3. 11

  4. 1.1

FAQs (Frequently Asked Questions)

1. What is a Perfect Cube?

Ans. 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 

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?

Ans. In Mathematics., 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 ∛2197 = ∛33

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.

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