Cube Root of 216

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Cube Roots

We know that to find the volume of the cube, we have volume = side3, but to find the side of a cube we have to take the cube root of the volume. 

The process of cubing is similar to squaring, only that the number is multiplied three times instead of two times as in squaring. The exponent used for cubes is 3, which is also denoted by the superscript³. Examples are 4³ = 4*4*4 = 64 or 8³ = 8*8*8 = 512 etc.

Thus, we can say that the cube root is the inverse operation of cubing a number. The cube root symbols is ∛, it is the “radical” symbol (used for square roots) with a little three to mean cube root.

The cube root of 216 is a value which is obtained by multiplying that number  three times It is expressed in the form of  ∛216 . The meaning of cube root is basically the root of a number which is generated by taking the cube of another number. Hence, if the value of  ∛216 = x, then x3 = 216 and we need to find here the value of x.

Cube root of 216( ∛216) = 6

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What is Cube Root?

The cube root of a number a is that number which when multiplied by itself three times gives the number ‘a’ itself. 

For Example, 23 = 8, or the cube root of 8 is 2

     33 = 27, or the cube root of 27 is 3

   43 = 64, or the cube root of 64 is 4

The symbol of the cube root is n3  or   ∛n

Thus, the cube root of 8 is represented as  ∛8 = 2 and that of 27 is represented as  ∛27 = 3 and so on.

We know that the cube of any number is found by multiplying that number three times. And the cube root of a number is the inverse operation of cubing a number. 

Example: If the cube of a number 53 = 125

Then cube root of ∛125 = 5

As 216 is a perfect cube, cube root of  ∛216 can be found in two  ways

Prime factorization method and Long Division method.


Calculation of Cube Root of 216

Let, ‘n’ be the value obtained from  ∛216, then as per the definition of cubes, n × n × n = n3 = 216. Since 216 is a perfect cube, we will use here the prime factorisation method, to get the cube root easily. Here are the following steps for the same.


Prime Factorisation Method

Step 1: Find the prime factors of 216

216 = 2 × 2 × 2 × 3 × 3 × 3

Step 2: 216 is a perfect cube. Therefore, group the factors of 216 in a pair of three and write in the form of cubes.

216 = (2 × 2 × 2) × (3 × 3 × 3)

216 = 23 × 33

Using the law of exponent, we get;

ambm = (ab)m

We get,

216 = 63

Step 3: Now, we will apply cube root on both the sides 

 ∛216 =  ∛63 = 6

Hence,  ∛216 = 6


Solved Examples

Example 1: Find the cube root of 512

Solution :

By Prime Factorisation method

Step 1: First we take the prime factors of a given number

512 = 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 

Step 2: Form groups of three similar factors

512 = 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2

Step 3: Take out one factor from each group and multiply.

= 23 x 23 x 23

= 83

Therefore, ∛512= 8

Example 2: Find the cube root of 1728

Solution :

By Prime Factorisation method

Step 1: First we take the prime factors of a given number 1728 

= 2 x 2 x 2 x 2 x 2 x 2 x 3 x 3 x 3

Step 2: Form groups of three similar factors

= 2 x 2 x 2 x 2 x2 x 2 x 3  x 3 x 3

Step 3: Take out one factor from each group and multiply.

= 23 x 23 x 33

= 123

Therefore,  ∛1728= 12


Quiz

  1. Using prime factorization, find the value of  ∛1331

  2. Using long division method, find the value of ∛729

FAQ (Frequently Asked Questions)

1. How to find cube root using Division Method?

Answer: For finding the cube root using the division method follow the following steps

  • Make a pair of 3 digit numbers from the back to front. 

  • Next  find the number whose cube root is less than or equal to the given number. 

  • Now, subtract the result from the given number and write down in the second number.

  •  Now find the multiplication factor for the further process in the long division method, which comes by multiplying the first number obtained. Similarly repeat the process, to find the cube root of a number. 

  • This method is used when the given number is not a perfect cube number. 

2. How to find cube root of Non-perfect Cubes?

Answer: We cannot find the cube root of numbers which are not perfect cube using the prime factorisation and estimation method. Hence, we will use here some other method.

Let us find the cube root of 110 here. Here, 110 is not a perfect cube.

Step 1

Now we would see 110 lies between 64 (cube of 4) and 125 (cube of 5). So, we will consider the lower number here, i.e. 4.

Step 2

Divide 110 by square of 4, i.e. 110/16 = 6.875 

Step 3

Now subtract 4 from 6.875 (whichever is greater) and divide it by 3. So,

6.875 - 4  = 2.875 & 2.875/3 = 0.95

Step 4

At the final step, we have to add the lower number which we got at the first step and the decimal number obtained.

So, 4 +0.95  = 4.95833

Therefore, the cube root of 110 is ∛110 = 4.9

This is not an accurate value but closer to it.