To rationalize the square root of 208, we follow these steps:

1. **Simplify the square root:**

\[ \sqrt{208} \]

2. **Factorize 208 into its prime factors:**

\[ 208 = 2 \times 104 \]
\[ 104 = 2 \times 52 \]
\[ 52 = 2 \times 26 \]
\[ 26 = 2 \times 13 \]

So,

\[ 208 = 2^4 \times 13 \]

3. **Simplify the square root using the prime factors:**

\[ \sqrt{208} = \sqrt{2^4 \times 13} \]

4. **Separate the factors under the square root:**

\[ \sqrt{208} = \sqrt{2^4} \times \sqrt{13} \]
\[ \sqrt{2^4} = 2^2 = 4 \]

Thus,

\[ \sqrt{208} = 4\sqrt{13} \]

The rationalized form of the square root of 208 is:

\[ \boxed{4\sqrt{13}} \]

Would you like more details or have any other questions?

Here are some related questions you might find interesting:

1. What is the square root of 50 rationalized?
2. How do you rationalize the denominator of \(\frac{1}{\sqrt{5}}\)?
3. Simplify the square root of 72.
4. How do you simplify the expression \(\sqrt{18} + \sqrt{32}\)?
5. What is the rationalized form of \(\frac{3}{\sqrt{12}}\)?

**Tip:** When simplifying square roots, always look for the largest perfect square factor to make the process easier.

Learn how to rationalize the square root of 208 with a step-by-step solution. Simplify the square root, factorize 208 into prime factors, and apply the principles of square root simplification. Suitable for high school students and those interested in mathematical operations on square roots.

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