The cube root of 2197, represented as ³√2197, is a value that results in the original number, getting multiplied by itself, three times. Basically, the cube root of number implies the root of a cubed number. In other words, if p³ = q, then ³√q = p. Therefore, the cube root is the inverse method of finding the cube of a number.

Let ³√2197 = x, then 2197 = x³. Now, we need to find here the value of x. The fastest method to find the value of the cube root of a four-digit number, 2197, is the estimation method. Although, we will also look into the prime factorisation method for better understanding.

Cube Root of 2197 By the Estimation Method

We need to memorise the cubes of natural numbers from 1 to 10, to proceed for the estimation method. Here is the table for the same.

Now let us proceed to calculate the value of ³√2197 following these steps:

Step 1:

Take the digit present at the unit’s place of 2197.

Unit Place of 2197 has —> 7

Step 2:

Check with the cubes table provided above. The cube of which number has 7 at its unit place?

Clearly, 3³ = 27

That means the cubic root of 2197 will have 3 at the unit place.

Say, ³√2197 = ?3

Step 3:

So, we have got the unit place digit of the required value.

Now ignore the last 3 digits of 2197, i.e. 197

Taking 2 as a reference number, we can see 2 lies between 1³ and 2³, i.e., between 1 and 8.

So, we will choose here the lowest number, nearest to 2, which is 1.

Since, 1³ = 1, therefore, we will have 1 as the first digit of the required number.

Hence, ³√2197 = 13

Cube Root of 2197 by Prime Factorisation Method

Here, we will use the prime factorisation method to find the prime factors of 2197. After that, we will pair the factors in a group of three to represent them as cubes. Since the cubes of a number cancel the cube root, therefore we will get the required value.

Let us understand step by step:

Step 1: Find the prime factors of 2197

If we try to divide 2197 by any prime number, we will come to know that the smallest prime number which can divide 2197 is 13. Therefore we get;

2197 = 13×13×13

Step 2: Pair the factors in a group of three and write in the form of cubes.

2197 = (13)¹⁺¹⁺¹ = (13)³ [By exponent law: a^mxaⁿ = a^m+n]

Step 3: Apply cube root to both sides of the above expression.

³√2197 = ³√(13)³

The cube root gets cancelled by the cube of 13.

Hence, ³√2197 =13

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Solved Examples

Q.1: What is the cube root of 9261?

Solution: By prime factorisation of 9261 we have;

9261 = 3 × 3 × 3 × 7 × 7 × 7

By grouping the products in a pair of three, we have;

9261 = (3 × 3 × 3) × (7 × 7 × 7) = 3³ × 7³

Taking cube root on both sides, we get;

³√9261 = ³√(3³ × 7³) = 3 × 7 = 21

Q.2: What is the cube root of 3375?

Solution: By prime factorisation of 3375 we have;

3375 = 5 × 5 × 5 × 3 × 3 × 3

By grouping the products in a pair of three, we have;

3375 = (5 × 5 × 5) × (3 × 3 × 3) = 5³ × 3³

Taking cube root on both sides, we get;

³√3375 = ³√(5³ × 3³) = 5 × 3 = 15