Cube Root by Prime Factorisation


 
 
Concept Explanation
 

Cube Root by Prime Factorisation

Procedure:

Step I: Obtain the natural number.

Step II: Express the given natural number as a product of prime factors.

Step III: Group the factors in triples in such a way that all the three factors in each triple are equal.  

Step IV: If no factor is left over in grouping in step III, then the number is a perfect cube, otherwise not.

 Step V: To find the natural whose cube is the given number, take one factor from each triple and multiply them. The cube of the number so obtained will be the given number.

Illustration 1: What is the smallest number by which 392 must be multiplied so that the product is a perfect cube?

Solution: Resolving 392 into prime factors, we get

               392= ({ 2times 2times 2 })times 7times 7

Grouping the factors in triplets of equal factors, we get

              392 =({2times 2times 2})times 7times 7

We find that 2 occurs as a prime factors of 392 thrice but 7 occurs as a prime factor only twice. Thus,if we multiply 392 by 7  ,7 will also occur as a prime factor thrice and the product will 2times 2times 2times 7times 7times 7, which is a perfect cube.

Hence, we must multiply 392 by 7 so that the product becomes a perfect cube.

Illustration 2: What is the smallest number by which 3087 must be divided so that the quotient is a perfect cube?

Solution: Resolving 3087 into prime factors, we get

3087 = 3times 3times 7times 7times 7

Grouping the factors in triplets of equal factors, we get

dpi{100} 3087 =3times 3times({7times 7times 7})

Clearly, if we divided 3087 by  3times 3= 9, the quotient would be 7times 7times 7 which is a perfect cube. Therefore, we must divide 3087 by 9 so that the quotient is a perfect cube.

Sample Questions
(More Questions for each concept available in Login)
Question : 1

The value of large (729;;X;;15625)^{1/3} is

Right Option : B
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Explanation
Question : 2

Which is the least number which when multiplied by 392, the product is the complete cube ?

Right Option : C
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Explanation
Question : 3

The value of sqrt[3]{frac{27}{64}} is

Right Option : A
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Explanation
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