Question

How do you write the prime factorization of \[64{n^3}\]?

Answer 1

Hint: Here, we will use the method of prime factorization to find the prime factors. We will first divide the given number by the least prime number. Then we will divide the result by either the same prime number or the next prime number and follow the same process until we get the quotient as a prime number. We will then multiply all the prime factors to get the required answer. Prime factorization is a method of finding factors of a number in terms of prime numbers.

Complete Step by Step Solution:
We are given with an expression \[64{n^3}\].
Now, we will find the prime factors of \[64\] by using the method of prime factorization.
As we can see 64 is an even number, so we will first divide it by the least prime number 2. Therefore, we get
\[64 \div 2 = 32\]
Again dividing 32 by 2, we get
\[32 \div 2 = 16\]
Now dividing 16 by 2, we get
\[16 \div 2 = 8\]
Dividing 8 by 2, we get
\[8 \div 2 = 4\]
Now dividing 4 by 2, we get
\[4 \div 2 = 2\]
As we got the quotient as a prime number, so we will not divide the number further .
Thus, the prime factors of the number \[64\] are \[2,2,2,2,2,2\].
We can write 64 as:
 \[64 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 = {2^6}\]
Now, we will assume that the variable \[n\] is a prime number.
Thus, the factors of \[{n^3}\] are \[n,n,n\] i.e., \[n \times n \times n = {n^3}\]

Therefore, the prime factors of \[64{n^3}\] by using the method of Prime factorization are \[{2^6} \times {n^3}\].

Note:
We know that the factor is defined as the whole number multiplied by a number to get another number. We should remember that we should use only the prime factors. We can also find the prime factors by using the factor tree method. In the factor tree method, the given number has to be multiplied by a prime factor and a composite factor. The composite factor has to be factorized by a prime factor and a composite factor. This has to be continued until all the factors become the prime factors. Each number is a factor of itself and the number 1 is a factor of every number. Thus, the number 1 can be neglected. The product should be the number itself and thus, the numbers become the prime factors.