Vedantu’s Cube Root of 9261 Explanation
The roots of a number are a very important topic in Mathematics. They form the basis of a large proportion of trigonometric and arithmetic calculations at higher-level mathematics. The root of a number is simply another number which if multiplied a certain time to itself will give the same number in return. For ex- 2 are the square root of 4 and 2 is also the cube root of 8. That is because if we multiply 2 by itself once we get 4 so we say the square of 2 is 4 and if we multiply 2 by 2 and 2 again then we get 8, which is said as 8 is the cube of 2. This can also be thought from the perspective of dimensions where every time a number is multiplied by itself, a new dimension is added. This same theory is used in the calculation and nomenclature of vector quantities in physics like velocity and acceleration.
How to find the cube root of 9261?
There are two methods to find the cube root of a number. They are:
The Estimation Method.
The Prime Factorization Method.
In this article, we will show the methods to find the cube root of 9261.
To calculate the cube root of 9261 by estimation method, it is important for us to learn the cubes of natural numbers from 1 to 9. These values are easy to learn and help the students to find the cube roots of any number and must be remembered by heart.
How to Determine Cube Root of 9261 by Estimation Method?
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 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.
9261−−−−√392613
= 21
Prime Factorization Method to Find the Cube Root of 9261
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:
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
9261−−−−√392613
= 21
FAQs on Cube Root of 9261
1. What does it mean to find the cube root of a number like 9261?
Finding the cube root of 9261 means figuring out which number, when multiplied by itself three times, equals 9261. It's the reverse operation of "cubing" a number. For example, the cube of 2 is 2 × 2 × 2 = 8, so the cube root of 8 is 2. The symbol for cube root is ∛. So, we are looking for the value of ∛9261.
2. What is the easiest method to find the cube root of 9261?
The most common method for finding the cube root of a number like 9261 is the prime factorization method. This involves:
- Breaking down 9261 into its prime factors.
- Grouping the identical factors into sets of three.
- Taking one factor from each group and multiplying them together to get the final answer.
For 9261, the prime factors are 3 × 3 × 3 × 7 × 7 × 7. This gives us one group of '3's and one group of '7's. So, the cube root is 3 × 7 = 21.
3. What is a perfect cube, and is 9261 an example of one?
A perfect cube is a whole number that is the result of multiplying another whole number by itself three times. For example, 27 is a perfect cube because 3 × 3 × 3 = 27. Yes, 9261 is a perfect cube because its cube root is a whole number, which is 21 (21 × 21 × 21 = 9261). If the cube root was a decimal, it would be a non-perfect cube.
4. How is finding a cube root different from prime factorization?
Prime factorization and finding a cube root are related but different processes. Prime factorization is the method of breaking down any number into a product of its prime numbers (e.g., 12 = 2 × 2 × 3). Finding the cube root is the goal, and prime factorization is often the tool used to achieve that goal. You use the prime factors to see if they can be grouped into identical sets of three to find the root.
5. Where are cube roots like the cube root of 9261 used in real life?
Cube roots are very important in geometry and science, especially when dealing with three-dimensional spaces. For example, if you know the volume of a perfect cube-shaped box is 9261 cubic centimetres, you can find the length of one of its sides by calculating the cube root. In this case, each side of the box would be 21 cm long.
6. Why do we need to group factors in threes when finding a cube root?
The word "cube" itself refers to the power of three. When we find a cube root, we are looking for a single number (let's call it 'x') that can be multiplied three times (x × x × x) to get our original number. By grouping the prime factors into sets of three, we are essentially reversing this process. Each group of three identical factors represents the 'x × x × x' that makes up the perfect cube. Taking one factor from each group gives us our 'x', the cube root.
7. Is the cube root of 9261 the same as its square root?
No, they are very different. The cube root of 9261 is the number that, when multiplied by itself three times, gives 9261. That number is 21. The square root of 9261 is the number that, when multiplied by itself just twice, gives 9261. The square root of 9261 is approximately 96.23, which is not a whole number. This shows that a number can be a perfect cube without being a perfect square.
8. How can you tell if a number might be a perfect cube just by looking at its last digit?
You can make a good guess by looking at the last digit of the number. Perfect cubes have a unique property where their last digit corresponds to the last digit of their cube root. For instance:
- A number ending in 1 will have a cube root ending in 1.
- A number ending in 8 will have a cube root ending in 2.
- A number ending in 7 will have a cube root ending in 3.
Since 9261 ends in 1, its cube root must also end in 1. This is a helpful trick for estimating cube roots quickly.
