Question

Using prime factorisation, find which of the following are perfect cubes

i) 2197 ii) 729 iii) 6859​

Answer 1

Answer:

1)Yes

2)Yes

3)Yes

Step-by-step explanation:

1} The process of finding the Prime Factors of 2197 is called Prime Factorization of 2197. To get the Prime Factors of 2197, you divide 2197 by the smallest prime number possible. Then you take the result from that and divide that by the smallest prime number. Repeat this process until you end up with 1.

This Prime Factorization process creates what we call the Prime Factor Tree of 2197. All the prime numbers that are used to divide in the Prime Factor Tree are the Prime Factors of 2197. Here is the math to illustrate:

2197 ÷ 13 = 169

169 ÷ 13 = 13

13 ÷ 13 = 1

2}Cube root of 729 by prime factorization method is 9

Solution:

To find cube root of 729 by prime factorization method

A number that must be multiplied times itself three times to equal a given number is called cube root

Prime factorization method:

Prime factorization is a number written as the product of all its prime factors.

In order of finding cube root by prime factorization we use the following steps:

Step I : Obtain the given number

Step II : Resolve it into prime factors.

Step III : Group the factors in 3 in such a way that each number of the group is same

Step IV : Take one factor from each group

Step V : Find the product of the factors obtained in step IV. This product is the required cube root

Prime factorization of 729:

729=3×3×3×3×3×3

Therefore

Cube root of 729=3×3

Thus cube root of 729 by prime factorization method is 9

3}We see that,

6859=19×19×19

Therefore

Cube root of 6859=19

Yes

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