HOW TO FIND THE CUBE ROOT OF A NUMBER USING PRIME FACTORIZATION

About "How to find the cube root of a number using prime factorization"

How to find the cube root of a number using prime factorization ?

Cube root through prime factorization method

Method of finding the cube root of a number

Step 1 : Resolve the given number into prime factors.

Step 2 : Write these factors in triplets such that all three factors in each triplet are equal.

Step 3 : From the product of all factors, take one from each triplet that gives the cube root of a number.

Example 1 :

Find the cube root of 512.

Solution :

512  =  2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2 

  =  (2 ⋅ 2 ⋅ 2) ⋅ (2 ⋅ 2 ⋅ 2) ⋅ (2 ⋅ 2 ⋅ 2)

Here we have nine 2's. We may group 2 as triples.

  =  2 ⋅ 2 ⋅ 2

  =  8

Hence cube root of 512 is 8.

Example 2 :

Find the cube root of 27 ⋅  64

Solution :

27 ⋅  64  =  ∛(3 ⋅ 3 ⋅ 3) ⋅ (4 ⋅ 4 ⋅ 4)

  =  3 ⋅ 4

  =  12

Hence the cube root of 27 ⋅  64 is 12.

Example 3 :

Find the cube root of 125/216

Solution :

∛125/216  =  ∛125 / ∛216

=  ∛(5 ⋅ 5 ⋅ 5) / (6 ⋅ 6 ⋅ 6)

  =  5 / 6

Hence the cube root of 125/216 is 5/6.

Example 4 :

Find the cube root of -512/1000

Solution :

∛(-512/1000)  =  ∛-512 / ∛1000

=  ∛(8 ⋅ 8 ⋅ 8) / (10 ⋅ 10 ⋅ 10)

  =  8 / 10

=  4/5

Hence the cube root of -512/1000 is 4/5.

Example 5 :

Find the cube root of 0.027

Solution :

∛(0.027 

First let us convert 0.027 as fraction, for that we have to multiply both numerator and denominator by 1000.

∛(0.027 =   ∛0.027(1000/1000)

=  ∛(27/1000)  

=  ∛27 / 1000

=  ∛(3 ⋅ 3 ⋅ 3) / ∛(10 ⋅ 10 10)

=  3 / 10

  =  0.3

Hence the cube root of 0.027 is 0.3

Example 6 :

Find the cube root of 12.25

Solution :

∛12.25

First let us convert 12.25 as fraction, for that we have to multiply both numerator and denominator by 100.

∛(12.25 =   ∛12.25(100/100)

=  ∛(1225/100)  

=  ∛1225 / 100

=  ∛(5 ⋅ 5 ⋅ 7⋅ 7) / ∛(10 ⋅ 10)

=  (5 ⋅ 7) / 10

  =  35/10

=  3.5

Hence the cube root of 12.25 is 3.5

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