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What is a Cube?

In this world, so many objects are three-dimensional which means that we can measure them on the basis of their length, breadth, and height. So yes, we can definitely say that 3-D figures are solid figures.

A cube which is a solid figure has all its sides equal if we take a measurement. And we can measure the quantities, the volume, or the capacity of an object with the help of cubic measurements such as cubic centimeter or cubic meter.

What is a Cube Root?

It is better if we start with an example before trying to understand its formal definition.

If we multiply 9 three times to itself, the product will be 729.

9 x 9 x 9 = 729. So, 9 will be called the cube root of 729. If you have understood the example, then let’s move on to its definition. A cube root is a number which when multiplied to itself thrice gives the product.

Symbol of a Cube Root

The symbol that we use to represent a cube root is the same as that of a square root with the only difference that in a square root, we use the number 2 and in cube root, we use the number 3. The root symbol can also be called a radical symbol. Here id how we represent a cube root:

\[\sqrt[3]{x}\]

If we break down 343 as 7 x 7 x 7, we can see that “7” is occurring thrice so it is the cube root of 343. We can also write it as \[\sqrt[3]{343}\] = 7

Prime Factorization of Perfect Cube

We can obtain a perfect cube or a cube number if we multiply a number to itself three times. For example, the prime factorization of 1728 will be 12 as 12 x 12 x 12 = 1728.

We can also check if a number is a perfect cube or not. For example, we want to see if 243 is a perfect cube or not? If we break down 243 as 3 x 3 x 3 x 3 x 3, we will see that it has five 3s and in a perfect cube, a group is made of three number that are equal. In this case, we can only make one group consisting of three 3s and we will be left with two extra 3s. Therefore, 243 is not a perfect cube.

Solved Examples

Example 1) Find the cube root of 1728

Solution 1) The first we do is find the prime factorization

So the prime factorization of 1728 = 2 x 2 x 2 x 2 x 2 x 2 x 3 x 3 x 3

= (2 x 2 x 3) x (2 x 2 x 3) x (2 x 2 x 3)

= 12 x 12 x 12

Therefore, 12 is the cube root of 1728

Example 2) Find the cube root of 9261

Solution 2) If we find out the factors of 9261, we will see that 3 x 3 x 3 x 7 x 7 x 7 are the factors of 9261.

Therefore, (7 x 3) x (7 x 3) x (7 x 3)

= 21 x 21 x 21

Therefore, 21 is the cube root of 9261

Example 3) Find the cube root of 15625

Solution 3) The factors of 15625 are 5 x 5 x 5 x 5 x 5 x 5

= (5 x 5) x (5 x 5) x (5 x 5)

= 25 x 25 x 25

Therefore, 25 is the cube root of 15625

Example 4) What will be the smallest number with which you can multiply 43904 to make it a perfect cube.

Solution 4) If we find out the prime factorization of 43904, our result will be:

2 x 2 x 2 x 2 x 2 x 2 x 2 x 7 x 7 x 7

So we can make two groups of 2s, each consisting of three 2s and next there is already a group consisting of three 7s. That is (2 x 2 x 2) x (2 x 2 x 2) x (7 x 7 x 7). In doing this, one 2 is left therefore in order to make a perfect cube, we need to more 2s i.e., it should be multiplied by 4

Example 5) What can be the smallest number by which 73002 be divided to make a perfect cube?

Solution 5) if we find the prime factorization of 73002, we will get 23 x 23 x 23 x 2 x 3.

Here, there is already a group of three 23s but 2 and 3 are left. So, if we divide the number by 6, a perfect cube can be achieved.

Example 6) Find the cube as well as the cube root of 27.

Solution 6) We can find the cube of 27 by multiplying it three times i.e., 27 x 27 x 27 = 19683. The prime factorization of 27 will be:

27 = 3 x 3 x 3

Therefore, the cube root of 27 is 3.

FAQ (Frequently Asked Questions)

Question 1: What is the Difference Between Square Root and Cube Root?

Answer: In a square root, we always multiply the number twice to itself whereas, in a cube root, we have to multiply a number thrice to itself. For example, consider the number 25. So 5 x 5 = 25. Here, 25 is the square of 5, and 5 is the square root of 25. Now let’s consider the number 7. So, 7 x 7 x 7 = 343. Here, 343 is the cube of 7, and 7 is the cube root of 343.

Question 2: How to Find Cube Root of 343 By Hand?

Answer: Yes we can find cube root of 343 by hand but there are a few steps that will make it easy for you.

Step 1: To find cube root of 343 or any number, first set up the problem in a proper format.

Step 2: Know the cube of every single number

Step 3: Think of a number that you can cube to produce the largest possible result but it should be less than than the first three numbers in the set.

Step 4: Copy the next three numbers from the set and then evaluate.

Step 5: For our first part of the divisor, whatever is on top of the radical sign, we have to write down three hundred times the square of it.

Step 6: Determine the rest of your divisors and do the same for the next.

Step 7: Multiply and subtract

Step 8: Write down your final answer.