Sure, let's go through the process of reducing the radical [tex]\(\sqrt{208}\)[/tex] step by step.
### Step 1: Identify Factors of 208
First, we need to list the factors of 208 to find the largest perfect square factor of 208.
Here are the factors of 208 in pairs:
- [tex]\(1 \times 208\)[/tex]
- [tex]\(2 \times 104\)[/tex]
- [tex]\(4 \times 52\)[/tex]
- [tex]\(8 \times 26\)[/tex]
- [tex]\(13 \times 16\)[/tex]
### Step 2: Identify Perfect Square Factors
Next, we want to identify which of these factors are perfect squares.
- [tex]\(1\)[/tex] is a perfect square [tex]\((1^2)\)[/tex]
- [tex]\(4\)[/tex] is a perfect square [tex]\((2^2)\)[/tex]
- [tex]\(16\)[/tex] is a perfect square [tex]\((4^2)\)[/tex]
So, the perfect square factors of 208 are [tex]\(1, 4,\)[/tex] and [tex]\(16\)[/tex].
### Step 3: Choose the Largest Perfect Square Factor
Among the perfect square factors ([tex]\(1, 4, 16\)[/tex]), the largest one is [tex]\(16\)[/tex].
### Step 4: Express as a Product of a Perfect Square and Another Factor
We can express [tex]\(208\)[/tex] as a product of the largest perfect square factor and another factor:
[tex]\[ 208 = 16 \times 13 \][/tex]
### Step 5: Simplify the Radical Using the Perfect Square
Now, we can use this factorization to simplify the radical:
[tex]\[ \sqrt{208} = \sqrt{16 \times 13} \][/tex]
### Step 6: Apply the Property of Radicals
We use the property of radicals that [tex]\(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\)[/tex], where [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are positive numbers.
[tex]\[ \sqrt{208} = \sqrt{16} \times \sqrt{13} \][/tex]
### Step 7: Simplify Further
Since [tex]\(\sqrt{16} = 4\)[/tex], we can simplify:
[tex]\[ \sqrt{208} = 4 \times \sqrt{13} \][/tex]
### Final Answer
So, the reduced form of the radical [tex]\(\sqrt{208}\)[/tex] is:
[tex]\[ \sqrt{208} = 4\sqrt{13} \][/tex]
This form shows the radical in its simplest terms with the largest possible perfect square factor extracted.
