\(\sqrt{2^{4}}+\frac{-2}{4 \cdot 3} \cdot \frac{2+4}{-8}: \frac{1}{2}\)

Let's break down and solve the expression step by step:

The expression is:

\[
\sqrt{2^{4}}+\frac{-2}{4 \cdot 3} \cdot \frac{2+4}{-8}: \frac{1}{2}
\]

### Step 1: Simplify \( \sqrt{2^{4}} \)

\[
\sqrt{2^{4}} = \sqrt{16} = 4
\]

### Step 2: Simplify \( \frac{-2}{4 \cdot 3} \)

First, calculate the denominator:

\[
4 \cdot 3 = 12
\]

Now, divide:

\[
\frac{-2}{12} = -\frac{1}{6}
\]

### Step 3: Simplify \( \frac{2+4}{-8} \)

First, add the numerator:

\[
2 + 4 = 6
\]

Now, divide:

\[
\frac{6}{-8} = -\frac{3}{4}
\]

### Step 4: Multiply the fractions

Now, multiply \(-\frac{1}{6}\) by \(-\frac{3}{4}\):

\[
-\frac{1}{6} \cdot -\frac{3}{4} = \frac{1 \times 3}{6 \times 4} = \frac{3}{24} = \frac{1}{8}
\]

### Step 5: Division by \(\frac{1}{2}\)

Now, divide \(\frac{1}{8}\) by \(\frac{1}{2}\):

\[
\frac{1}{8} : \frac{1}{2} = \frac{1}{8} \cdot \frac{2}{1} = \frac{2}{8} = \frac{1}{4}
\]

### Step 6: Add the results

Finally, add the results from steps 1 and 5:

\[
4 + \frac{1}{4} = \frac{16}{4} + \frac{1}{4} = \frac{17}{4}
\]

Thus, the final answer is:

\[
\boxed{\frac{17}{4}}
\]

Would you like more details on any step or have any questions? 

Here are 5 related questions to explore further:

1. How do you simplify expressions involving square roots?
2. What are the rules for multiplying and dividing fractions?
3. How do you handle negative numbers in fractions?
4. What is the importance of order of operations in mathematical expressions?
5. How do you convert improper fractions to mixed numbers?

**Tip:** When simplifying expressions, always remember to follow the order of operations (PEMDAS: Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)).

Learn how to simplify expressions involving square roots and fractions step-by-step. This detailed guide covers basic fraction operations and the order of operations (PEMDAS). Suitable for students in Grades 6-8.

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