Write 243 as a product of prime factors and express it in exponential form.

To express the number 243 as a product of its prime factors and in exponential form, we first need to break it down into its prime components. Prime factorization involves dividing the number by the smallest prime numbers until we reach 1. Let's walk through the steps together.

Step-by-Step Prime Factorization

We start with the number 243. The first thing to notice is that 243 is an odd number, so it cannot be divided by 2. The next smallest prime number is 3.

Dividing by 3

Let's divide 243 by 3:

• 243 ÷ 3 = 81

Now we have 81. We can continue dividing by 3:

• 81 ÷ 3 = 27

Next, we take 27:

• 27 ÷ 3 = 9

Continuing with 9:

• 9 ÷ 3 = 3

Finally, we divide 3 by itself:

• 3 ÷ 3 = 1

Collecting the Prime Factors

Now, let's summarize the divisions we performed:

• 243 = 3 × 81
• 81 = 3 × 27
• 27 = 3 × 9
• 9 = 3 × 3

From this, we can see that 243 can be expressed as:

• 243 = 3 × 3 × 3 × 3 × 3 = 3⁵

Final Expression in Exponential Form

Thus, the prime factorization of 243 in exponential form is:

243 = 3⁵

This means that 243 is made up of the prime number 3 multiplied by itself five times. Understanding prime factorization is essential in various areas of mathematics, including simplifying fractions, finding least common multiples, and solving problems related to divisibility.