Let's solve the expression step by step:

Given:
\[
54 + \left[ 19 - \{ 15 - (6 - \frac{16}{4} \times 8) \} \right]
\]

### Step 1: Simplify inside the innermost parentheses
Start by solving the division and multiplication inside the parentheses:

\[
\frac{16}{4} = 4
\]

\[
4 \times 8 = 32
\]

So, the expression inside becomes:
\[
6 - 32 = -26
\]

### Step 2: Substitute back into the expression
Substitute \(-26\) back into the original expression:

\[
54 + \left[ 19 - \{ 15 - (-26) \} \right]
\]

### Step 3: Simplify inside the braces
Next, solve the expression inside the braces:

\[
15 - (-26) = 15 + 26 = 41
\]

So the expression now becomes:
\[
54 + \left[ 19 - 41 \right]
\]

### Step 4: Simplify inside the brackets
Now, simplify inside the brackets:

\[
19 - 41 = -22
\]

### Step 5: Final calculation
Finally, add the result to 54:

\[
54 + (-22) = 54 - 22 = 32
\]

So, the final answer is:
\[
\boxed{32}
\]

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

Here are 5 related questions to expand your understanding:
1. What is the order of operations (PEMDAS) and how does it apply to solving expressions?
2. How do parentheses and brackets affect the evaluation of an expression?
3. Can you solve the expression \(54 + \left[19 - \{15 + (6 + 2 \times 4)\}\right]\)?
4. What happens if there are nested parentheses within an expression?
5. How does changing the position of operations in an expression affect the result?

**Tip:** Always perform operations inside the innermost parentheses first to avoid mistakes in simplifying complex expressions.

Learn how to solve the arithmetic expression 54 + [19 - {15 - (6 - 16 ÷ 4 × 8)}] step-by-step. This problem involves applying the order of operations, handling parentheses and brackets in arithmetic, and rules for division and multiplication. Suitable for students in Grades 6-8.

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