What is the prime factorization of 64?

Understand the Problem

The question is asking for the prime factorization of the number 64, which involves breaking it down into its prime factors.

The prime factorization of $64$ is $2^6$.

The prime factorization of $64$ is $2^6$.

Steps to Solve

1. Identify the number for factorization

Start with the number $64$ that we want to factor.

1. Divide by the smallest prime number

Check if $64$ is divisible by the smallest prime number, which is $2$:

$$64 \div 2 = 32$$

1. Continue dividing by $2$

Repeat the division with $2$ until you can no longer divide evenly:

$$32 \div 2 = 16$$
$$16 \div 2 = 8$$
$$8 \div 2 = 4$$
$$4 \div 2 = 2$$
$$2 \div 2 = 1$$

1. List the prime factors

Each time we divided by $2$, we obtained the prime factor. Since we divided by $2$ a total of $6$ times, we can express this as:

$$64 = 2 \times 2 \times 2 \times 2 \times 2 \times 2$$

1. Write the prime factorization in exponential form

Using exponents, we can simplify the prime factorization:

$$64 = 2^6$$

The prime factorization of $64$ is $2^6$.

The number $64$ can also be expressed in different ways. It is $8^2$ (as $8$ is $2^3$), which is another interesting way to view it in terms of powers.
Some common mistakes when finding prime factorizations include forgetting to check divisibility by all applicable prime numbers or stopping too early in the division process. Always continue dividing until you reach $1$.