Let's simplify and solve the given expression:

\[
\sqrt{8 + 3(2 + 6)} \quad \text{where the square root is applied to the entire expression.}
\]

### Step-by-step:

1. **Simplify inside the parentheses**:
   \[
   2 + 6 = 8
   \]
   So the expression becomes:
   \[
   \sqrt{8 + 3(8)}
   \]

2. **Multiply 3 by 8**:
   \[
   3 \times 8 = 24
   \]
   Now the expression is:
   \[
   \sqrt{8 + 24}
   \]

3. **Add 8 and 24**:
   \[
   8 + 24 = 32
   \]
   Now we have:
   \[
   \sqrt{32}
   \]

4. **Simplify the square root of 32**:
   \[
   \sqrt{32} = \sqrt{16 \times 2} = \sqrt{16} \times \sqrt{2} = 4\sqrt{2}
   \]

Thus, the simplified result is:
\[
4\sqrt{2}
\]

Would you like further explanation or clarification? Let me know!

### Related Questions:
1. How can we simplify other square roots with products, like \(\sqrt{18}\)?
2. What is the process for simplifying square roots in algebraic expressions?
3. How do you apply the distributive property when simplifying expressions?
4. How do you calculate \(\sqrt{50}\) using prime factorization?
5. How do we handle square roots in equations with variables?

### Tip:
When simplifying square roots, always try to factor out perfect squares (like 16, 25, 36, etc.) to make the process easier!

The query submitted by the user: sqrt(8 + 3(2 + 6))

Learn how to simplify expressions involving square roots and multiplication, such as sqrt(8 + 3(2 + 6)), with a step-by-step breakdown. This solution covers key algebraic concepts including the distributive property and properties of square roots. Suitable for middle school students.

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