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The cube of any number is the number that we get when a number is multiplied by itself twice. The cube root, on the other hand, is the number which, when multiplied by itself twice gives the number whose cube root is to be found as the product. Cubes and cube roots are Mathematically inverse concerning each other. The cube of any natural number is another natural number itself. These cubes of natural numbers are called perfect cubes. The cube root of perfect cube numbers is always a natural number only.

The cube root of 9261 is that number which when multiplied by itself twice gives 9261 as the answer. The number that gives 9261 as the product when multiplied by itself twice is 21.

21 x 21 x 21 = 441 x 21 = 9261

There are multiple ways to arrive at an answer to this question. Either of the two methods namely can determine the cube root of any number which is a perfect cube:

Prime factorization Method

Estimation method

If the number is not a perfect cube, then the cube root cannot be determined using any of these two methods. So, to find the cube root of a number by prime factorization or estimation method, we should first check whether the given number is a perfect cube or not.

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 is 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.

So, to find the cube root of 9261, we should divide the number into groups of three digits starting from the digit in the unit’s place.

009 261

Step 2:

From the first group starting from the right, note down the unit’s digit.

The first rightmost group in step 1 is 261, and the digit in its unit’s place is 1.

Step 3:

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

The digit in the unit’s place obtained in step 1 is ‘1’ and hence the unit’s digit of the cube root of 9261 is also ‘1’.

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 A3 and B3 and is more closer to B3, then the ten’s digit of its cube root is considered as B.

In the given number 9261, the second group of 3 digits from the right is 009.

This number lies between two perfect cube numbers 8 and 27 i.e. 23 and 33. Because 9 is more close to 8 i.e. 23, the ten’s digit of the cube root of 9261 is 2.

The cube root of 9261 found using the estimation method is 21.

\[\sqrt[3]{9261}\] = 21

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 resolved into its prime factors. It is 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.

The given number whose cube root is to be determined is 9261. The lowest prime number is ‘2’ and 9261 is not divisible by 2. So, let us go with the next lowest prime number i.e. 3.

9261 when divided by 3 gives 3087 as the answer.

3087 is further divisible by 3. 3087 when divided by 3 gives 1029.

1029 is still divisible by 3. 1029 divided by 3 gives 343.

The number 343 is not divisible by 3 and hence with go with the next lowest prime number. The next lowest prime number is 5. But 343 is not divisible by 5. The next prime number in the queue is 7 and 343 is divisible by 7.

343 divided by 7 gives 49 and 49 divided by 7 is 7. This complete sequence of steps can be summarized as:

(Image to be added soon)

Step 2:

Write the number whose cube root is to be determined as the product of their primes.

The number 9261 can be written as the product of its primes as:

9261 = 3 x 3 x 3 x 7 x 7 x 7

Step 3:

Divide the factors into groups consisting of three identical factors.

9261 as the product of its primes is rewritten as:

9261 = 3 x 3 x 3 x 7 x 7 x 7

Step 4:

The cube root of the number is found as the product of one factor taken from each group in step 3.

So, the cube root of 9261 is found to be 3 x 7 = 21

\[\sqrt[3]{9261}\] = 21

The cube of the cube root of 9261 is 9261 itself.

Prime factorization method can be used to check whether a given number is a perfect cube number or not and if the number is not a perfect cube, then this method can be used to find the smallest number to be multiplied or divided by the given number to make it a perfect cube number.

FAQ (Frequently Asked Questions)

1. What is Cube and Cube Root?

Cube and cube root are the most important concepts in various mathematical calculations. A Cube of any number is that number which we get when we multiply the number whose cube is to be determined by itself twice. In other words, the cube of a number is the number raised to the power 3. If the number is ‘p’, then its cube is p x p x p or p^{3}. Cube root is the inverse operation of a cube. The Cube root of a number is that number which when multiplied by itself twice gives the number whose cube root is to be determined as the answer.

2. How Do You Find the Cube Root of 9261?

Cube root of 9261 is that number which when multiplied by itself two times gives 9261 as the answer. The Cube root of 9261 is symbolically represented as ∛9261 and can be exponentially written as (9261)^{1/3}. The cube root of 9261 can be determined by any one of the following ways:

Check the lookup table for cube of first 25 natural numbers

Prime factorization method

Estimation method

The cube root of 9261 found using any of these methods is 21.