To simplify $\sqrt{164}$, we can follow these steps:

1. Factorize 164 into its prime factors: $164 = 2 \times 2 \times 41 = 2^2 \times 41$

2. Apply the square root property: $\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}$

   Thus: $\sqrt{164} = \sqrt{2^2 \cdot 41} = \sqrt{2^2} \cdot \sqrt{41}$

3. Simplify the perfect square: $\sqrt{2^2} = 2$

   Therefore: $\sqrt{164} = 2\sqrt{41}$

4. Approximate the value of $\sqrt{41}$: $\sqrt{41} \approx 6.403$

   So: $\sqrt{164} \approx 2 \cdot 6.403 = 12.806$

Final Answer:

$\sqrt{164} = 2\sqrt{41} \quad \text{or approximately } 12.806$

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Would you like further explanation or details?
Here are 5 related questions to deepen your understanding:

1. How do you identify the prime factorization of a number?
2. What is the general rule for simplifying square roots?
3. How can you estimate square roots of non-perfect squares?
4. What is the significance of expressing roots in simplest radical form?
5. How does the square root function behave for large versus small numbers?

Tip: When simplifying square roots, always look for the largest perfect square factor!