Cube & Cube Roots: Theory and Solved Examples

Before learning how to find the cube root, let us first learn the meaning of cube. The process of cubing a number is multiplying the number three times. The exponent used for cubes is 3, which is also denoted by the x³. For example:  The cube of 4 will be calculated as 4 * 4 * 4 = 4³ = 64 or 8³ = 8 * 8 * 8 = 512.

Now in order to find the cube root of a number the only method available is prime factorization. The approach applied is that each of the number, which is a perfect cube, will have every prime factor appearing in a group of 3. This is done because unlike square root there is no other conventional method to find the cube root.  After the prime factorization, each of the prime factors is selected once for every three times it is appearing in the number. You will understand the concept better with the help of the following examples.

 

Example 1:  Find out the cube root of 1728.

Sol: First we will do prime factorization.

Prime factorization of 1728 is = 2 * 2 * 2 * 2 * 2 * 2 * 3 * 3 * 3 
= (2 * 2 * 3) * (2 * 2 * 3) * (2 * 2 * 3).  
= 12 * 12 * 12 the cube root of 1728 is 12.

 

Example 2: Find the cube root of 9261.

Sol: The factors of 9261 are 3 * 3 * 3 * 7 * 7 * 7.

= (7 * 3) * (7 * 3) * (7 × 3) 
= 21 * 21 * 21 Þ the cube root of 9261 is 21.

 

Example 3: Find the cube root of 15625.

Sol: The factors of 15625 are = 5 * 5 * 5 * 5 * 5 * 5.

= (5 * 5) * (5 * 5) * (5 * 5)
=25 * 25 * 25 Þ the cube root of 15625 is 25.

 

Example 4: Find the smallest number with which 43904 should be multiplied so as to make it a perfect cube.

Sol: The prime factorization of 43904 is 2 * 2 * 2 * 2 * 2 * 2 * 2 * 7 * 7 * 7. As you can see 2 appears in this 7 times, so groups of three 2s can be made 2 times and still one 2 will be left and 7 appears in a group of three only. Thus the groups can be made as 2 * 2 * 2 * 2 * 2 * 2 * 2 * 7 * 7 * 7 and as you see a single 2 is left out and now it needs two more 2s to make it a perfect cube i.e. it should be multiplied by 4.

 

Example 5: Find the smallest number by which 73002 should be divided so as to make it a perfect cube.

Sol: The prime factorization of the number 73002 is 23 * 23 * 23 * 2 * 3. As you can see 23 already appears three times, there is a single 2 and a single 3. This implies if this number is divided by 6, it will become a perfect cube. Hence the answer to the question will be 6.

 

Example 6: Find the cube and cube root of 27.

Sol: The cube of 27 will be found by multiplying it three times i.e. 27 * 27 * 27 = 19683.

Now the prime factorization of 27 is 3 * 3 * 3. As you know, you need to take one number for every group of three. As 3 appear three times, taking a single 3 for the same, the cube root of 27 is 3.