Cube Root of 512: Explained with Simple Steps

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 is also known as the radical symbol. Here id how we represent a cube root:                              

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

The Cube Root Of 512

The Cube Root of 512 is 8

If we break down 512 as 8 x 8 x 8, we can see that “8” is occurring thrice so it is a the cube root of 512. We can also write it as \[\sqrt[3]{512}\] = 8

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 numbers 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 Example 

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 as Well as the Cube Root of 27. 

Solution 2) 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.   

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 Can Be the Smallest Number by Which 73002 Be Divided to Make a Perfect Cube?

Solution 4) 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 5) What Will Be the Smallest Number With Which You Can Multiply 43904 to Make it a Perfect Cube. 

Solution 5)  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 6) Find the Cube Root of 9261

Solution 6)  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