Question

Find the cube root of the following number by prime factorization method: 91125
a) 45
b) 35
c) 55
d) 465

Answer 1

Hint: Prime factorization of any number means writing the given number in the form of products of prime numbers. A cube root can be found with the help of making groups of 3 same numbers and just finding the product of grouped numbers at once.

Complete step-by-step answer:
Prime factorization of any number means that it splits the given number into a product of prime numbers. Prime numbers are the numbers which have only two factors 1 and the number itself and it is defined for only natural numbers greater than 1.

And we know the cube root with the help of prime factorization can be given by multiplying the pairs of 3 numbers taken at once. So, we have the number 91125 of which we need to find the cube root with the help of prime factorization as discussed above. So, let us factorize the given number in terms of prime numbers as

So, we can express 91125 in terms of the prime numbers as follows:
$91125=3\times 3\times 3\times 3\times 3\times 3\times 5\times 5\times 5$
Now, make the group of 3 same prime numbers from the above factorizations. So, we can observe that there are two groups of number ‘3’ , one group of ‘5’. 2 pairs of number ‘3’ and one pair of number ‘5’, if we make three of the same numbers. Hence, we can rewrite the factors of 91125. This can be done by taking one number and multiplying.
Now, take one number from each number and multiply them to get the cube root of the number 91125. Hence, cube root of $91125=3\times 3\times 5=45$.
So, option (a) is correct.

Note: Remember that we need to use only prime numbers for dividing the given numbers i.e. we need to represent the given number in factorization form only with the help of prime numbers. As one may write the factorization as $91125=9\times 15\times 3\times 3\times 3\times 5\times 5$ , which is wrong as we will not be able to make a group of three same numbers. So, take care and use prime numbers only.
We can use only the order of division of the number by the prime numbers. One may divide the number by ‘5’ in the starting of division as well. So, one may proceed as

It means we can use prime numbers for division in any order. But be cleat that uses only prime numbers for division. So we get the value of ${{\log }_{10}}10$ from the above relation as ‘1’. Hence, the value of M will be ‘1’.