To create a prime factor tree for 200, we start by finding the smallest prime number that divides 200. The smallest prime number is 2.

• 200 ÷ 2 = 100
• 100 ÷ 2 = 50
• 50 ÷ 2 = 25
• 25 ÷ 5 = 5
• 5 ÷ 5 = 1

The prime factorization of 200 is 2 × 2 × 2 × 5 × 5, or 2³ × 5².

The prime factor tree for 200 can be visualized as follows:

• 200
  • 2
    • 100
      • 2
        • 50
          • 2
            • 25
              • 5
                • 5
                • 1

Creating a prime factor tree is a systematic way to break down a number into its prime factors. Let's take a closer look at how we can construct a prime factor tree for the number 200, step by step.

Starting with the Basics

To begin, we need to identify the smallest prime number that can divide 200. The smallest prime number is 2. We can divide 200 by 2, which gives us:

• 200 ÷ 2 = 100

Continuing the Division

Next, we take the quotient, which is 100, and divide it by 2 again:

• 100 ÷ 2 = 50

We repeat this process with 50:

• 50 ÷ 2 = 25

Switching to the Next Prime

At this point, we can no longer divide 25 by 2 since it is not an even number. Instead, we move to the next smallest prime number, which is 5:

• 25 ÷ 5 = 5

Finally, we divide 5 by itself:

• 5 ÷ 5 = 1

Constructing the Prime Factor Tree

Now that we have reached 1, we can summarize our findings. The prime factorization of 200 can be expressed as:

• 2 × 2 × 2 × 5 × 5

This can also be written in exponential form as:

• 2³ × 5²

Visual Representation

To visualize this process, we can draw a prime factor tree:

• 200
• 2
• 100
• 2
• 50
• 2
• 25
• 5
• 5
• 1

Understanding Prime Factorization

Prime factorization is essential in various areas of mathematics, including simplifying fractions, finding the least common multiple (LCM), and greatest common divisor (GCD). By breaking down numbers into their prime factors, we can better understand their properties and relationships.

In summary, the prime factor tree for 200 illustrates how we can systematically break down a composite number into its prime components, leading us to the conclusion that:

• The prime factorization of 200 is 2³ × 5².