**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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