Question

What is the exponent of 3 in the prime factorization of 864?

Answer 1

Hint: In the question, first we will prime factorize 864. Here we will try to find the prime factors of 864. After that write the exponential form of the factorization of that number. When we multiply these prime factors together then we will get the actual number on which prime factorization is done. Let’s see how we can solve it further.

Complete step-by-step answer:
 Prime factorization is a process of factoring a number in terms of prime number. The simplest algorithm to find the prime factors at a number is to keep on dividing the original number by prime factors until we get the remainder equal to 1.
The prime factorization of 864 = 2 $ \times $ 2 $ \times $ 2 $ \times $ 2 $ \times $ 2 $ \times $ 3 $ \times $ 3 $ \times $ 3
We will write this in exponential form of 864, that is;
 $ \Rightarrow $ 864 = 2 $ \times $ 2 $ \times $ 2 $ \times $ 2 $ \times $ 2 $ \times $ 3 $ \times $ 3 $ \times $ 3 = $ {2^5} \times {3^3} $
Therefore, the exponent of 3 in the prime factorization of 864 is 3.

Note: We will see some more examples of prime factorization:
12 = 2 $ \times $ 2 $ \times $ 3 $ \times $ 1
24 = 2 $ \times $ 2 $ \times $ 2 $ \times $ 3 $ \times $ 1
49 = 7 $ \times $ 7 $ \times $ 1
63 = 3 $ \times $ 3 $ \times $ 7 $ \times $ 1
21 = 3 $ \times $ 7 $ \times $ 1
11 = 11 $ \times $ 1
Those numbers which have only two prime factors that is 1 and the number itself, they are called prime numbers. For example: 11, 13, 17, 19, 23 etc,.