How to Calculate the Cube Root of 2197: Step-by-Step Guide
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.
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.
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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.
Let us now calculate the cube root of 2197 by using the estimation method in a step-by-step manner.
Consider the digit given at the unit’s place of 2197.
The unit place 2197 has 7.
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.
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.
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
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)
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:
Let us consider the last 3 digits of 4096 as the first part and the remaining digit 4 as the second part.
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.
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:
First, we will find the prime factors of 343.
So, the prime factors of 343 are 7 × 7 × 7.
Now, we will pair similar factors in a group and represent them as cubes.
343 = (7 * 7 * 7)
343 = 73
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
What will be the cube root of .000216?
6
.06
77
87
Ans: b
Find the number which is not a perfect cube among the numbers given below.
1331
216
243
512
Ans: c
What will be the cube root of 1.331?
0.11
0.011
11
1.1
Ans: d
FAQs on Cube Root of 2197 Explained with Examples
1. What is the value of the cube root of 2197?
The cube root of 2197 is the number that, when multiplied by itself three times, equals 2197. The value is 13, because 13 × 13 × 13 = 2197. In mathematical notation, this is written as ∛2197 = 13.
2. How is the cube root of 2197 found using the prime factorization method?
To find the cube root of 2197 with the prime factorization method, you follow these steps as per the CBSE syllabus:
- First, break down 2197 into its prime factors. The number 2197 is not divisible by smaller primes like 2, 3, 5, 7, or 11.
- Upon testing with the next prime number, 13, we find that 2197 ÷ 13 = 169.
- We know that 169 is a perfect square: 169 = 13 × 13.
- Therefore, the complete prime factorization of 2197 is 13 × 13 × 13.
- To find the cube root, we group these identical factors into a set of three (13³). For each group, we take one factor out.
- This gives us the cube root, which is 13.
3. Is 2197 a perfect cube? How can you tell?
Yes, 2197 is a perfect cube. A number is classified as a perfect cube if its cube root is a whole number (an integer). Since the prime factorization of 2197 results in a single prime factor (13) repeated exactly three times (13³), its cube root is the integer 13. This confirms that 2197 is a perfect cube.
4. How can estimation be used to find the cube root of 2197?
Estimation is a quick way to narrow down the possible answer. You can start with cubes you already know:
- 10³ = 1000
- 20³ = 8000
Since 2197 is between 1000 and 8000, its cube root must be between 10 and 20. Additionally, look at the last digit of 2197, which is 7. The only single-digit number whose cube ends in 7 is 3 (since 3³ = 27). This strongly suggests the cube root must be a number ending in 3, such as 13. A quick check (13 x 13 x 13) confirms the answer is 13.
5. What is the key difference between finding the cube of 13 and the cube root of 2197?
Finding the cube and the cube root are inverse mathematical operations:
- Finding the cube of 13 involves multiplication: 13³ = 13 × 13 × 13, which equals 2197. You are starting with the base number and calculating its cubed value.
- Finding the cube root of 2197 involves finding the original base number: ∛2197. You are starting with the cubed value and working backwards to find the number that, when multiplied by itself three times, results in 2197.
6. Why is the prime factorization method so effective for determining cube roots of integers?
The prime factorization method is effective because it breaks a number down into its most fundamental components (its prime factors). For a number to be a perfect cube, all of its prime factors must exist in groups of three. This method provides a clear and visual way to check for these triplets. If all prime factors can be bundled into groups of three, the number is a perfect cube, and its root can be found easily by taking one factor from each group. This removes guesswork and provides a definitive proof.
7. Where might you apply the concept of cube roots in a real-world scenario?
The concept of cube roots is fundamentally important in problems related to volume. For example, if an engineer needs to design a cubic water tank that holds exactly 2197 cubic metres of water, they would need to find the length of each side of the tank. By calculating the cube root of the volume (∛2197), they would determine that each side of the tank must be 13 metres long. This application is crucial in fields like architecture, physics, and engineering.
