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Cube Roots

If the cube of 2 is 8, then 2 is said to be the cube root of 8. We denote cube root by the symbol
Thus

Cube root : The cube root of a given number is that number whose cube is equal to the given number. 

Cube Root of a Perfect Cube by Prime Factorisation
We have seen that a number is a perfect cube if it can be expressed as the product of triples of equal factors.
The following example shows how this can be used to find the cube root of a perfect cube.

 

Example

Find the cube root of 1728.

Solution

1. Carry out the prime factorisation of the given number.

1728 = 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 × 3

2. Make triples of equal factors.

(2 × 2 × 2) × (2 × 2 × 2) × (3 × 3 × 3)

3. Take one prime factor from each triple and find their product.
2 × 2 × 3 = 12

12 is the required cube root, i.e.

 

 
Example

Find the cube root of 729

Solution

729 = 3 × 3 × 3 × 3 × 3 × 3.

729 = 33× 33


Cube Root of a Negative Number :-
We have seen that the cube of a negative integer is negative.

 

Example : (- 5)3 = (- 5) x (- 5) x (- 5) = (- 125)


It follows that = (- 5)
 

Example : = - = (- 5)

In general, if x is a positive integer,   

 

Thus, to find the cube root of a negative integer, find the cube root of its absolute value, and put a negative sign before it.
 

Example :   

Cube root of a perfect cube without using prime factorization.


We can estimate cube root of given number.

Consider the number 19683.


1. Split the number from the right into group of three digit numbers.

    19683.

    We have two parts 19 and 683.

2. The unit digit of 683 is 3. Hence the unit digit of required cube root must be 7.


3. We have to estimate the cube root of 19.

    8 < 19 < 27.

   < <

   2 < < 3.


4. Take unit digit of the smallest number.

The cube root of 19 lies between 2 and 3. It is ‘2’.

So, the cube root of 19683 is 27.

 

Example

Estimate the cube root of 110592

Solution

i.110 592

ii.The unit digit of 592 is 2. So, the unit digit of required cube root is 8.

iii.Now, 64 < 110 < 125.

    < <

    4 < < 5

iv.The smallest number is 4.

v.Therefore, the cube root of 110592 is 48.


For your Knowledge :-

Cube Root of a Perfect Cube Expressed in Exponential Form
Consider the number 13824. In the exponential form it can be written as 13824 = 29× 33
If the indices of the factors are multiplies of 3, then the number is a perfect cube. Even if one factor has an index which is not a multiple of 3, then it is not a perfect cube.

 

Example : 6912 = 28× 33 is not a perfect cube.


The cube root is obtained by dividing each index by 3. 

Thus  
Cube Root of Product of Integers
 

Example

Solution

64 = 2 × 2 × 2 × 2 × 2 × 2

125 = 5 × 5 × 5


Notice also that,


Cube Root of a Rational Number :-

We have seen earlier that

It follows from this that

 

Example

Find the cube root of

Solution





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