Square Root Of 208 In Radical Form

Simplifying the Square Root of 208: A Step-by-Step Guide

Finding the square root of 208 in its simplest radical form might seem daunting at first, but with a methodical approach, it's surprisingly straightforward. This guide will walk you through the process, explaining each step so you can confidently tackle similar problems. This article covers simplifying radicals, prime factorization, and perfect squares, providing a comprehensive understanding of square root simplification.

What is a radical form? A radical form represents a number using a radical symbol (√), indicating a root (in this case, a square root). Simplifying a radical means expressing it without any perfect square factors under the radical sign.

Understanding Prime Factorization

The key to simplifying square roots lies in prime factorization. Prime factorization is the process of breaking down a number into its prime factors – numbers only divisible by one and themselves. Let's find the prime factorization of 208:

• 208 is divisible by 2: 208 = 2 × 104
• 104 is divisible by 2: 104 = 2 × 52
• 52 is divisible by 2: 52 = 2 × 26
• 26 is divisible by 2: 26 = 2 × 13
• 13 is a prime number:

Therefore, the prime factorization of 208 is 2 × 2 × 2 × 2 × 13, or 2⁴ × 13.

Identifying Perfect Squares

Now that we have the prime factorization, we look for perfect squares within the factors. A perfect square is a number that results from squaring another whole number (e.g., 4 is a perfect square because 2² = 4). In our prime factorization of 208 (2⁴ × 13), we have 2⁴, which is (2²)². This means we have a perfect square: 2² x 2² = 16.

Simplifying the Radical

We can now rewrite the square root of 208 using the perfect square we identified:

√208 = √(2⁴ × 13) = √(2² × 2² × 13) = √(16 × 13)

Since √(a × b) = √a × √b, we can separate the perfect square:

√(16 × 13) = √16 × √13

Finally, we simplify:

√16 = 4

So, the simplified radical form of √208 is 4√13.

Summary of Steps to Simplify Radicals

To summarize, here's the general process for simplifying square roots:

1. Find the prime factorization: Break down the number under the radical into its prime factors.
2. Identify perfect squares: Look for pairs of identical prime factors. Each pair represents a perfect square.
3. Simplify: Take the square root of each perfect square and move it outside the radical symbol. Leave any remaining factors under the radical.

By following these steps, you can confidently simplify any square root and express it in its simplest radical form. Remember to practice – the more you work through these problems, the more comfortable you’ll become with simplifying radicals.