Let's break down the given expression step by step:

Expression: $7 - \left\{ 5 + 10 \left[ \frac{20}{5} - 2 + 4 \times (5 + 2 \times 3) \right] - 8 \times 3^2 \right\} + 50 \times (6 \times 2)$

Step 1: Solve the innermost parentheses

First, handle $2 \times 3$ inside the parentheses:

$2 \times 3 = 6$

Now the expression becomes: $7 - \left\{ 5 + 10 \left[ \frac{20}{5} - 2 + 4 \times (5 + 6) \right] - 8 \times 3^2 \right\} + 50 \times (6 \times 2)$

Step 2: Simplify the next parentheses

Next, calculate $5 + 6$:

$5 + 6 = 11$

So now we have: $7 - \left\{ 5 + 10 \left[ \frac{20}{5} - 2 + 4 \times 11 \right] - 8 \times 3^2 \right\} + 50 \times (6 \times 2)$

Step 3: Handle the multiplication inside the brackets

Now compute $4 \times 11$:

$4 \times 11 = 44$

This turns the expression into: $7 - \left\{ 5 + 10 \left[ \frac{20}{5} - 2 + 44 \right] - 8 \times 3^2 \right\} + 50 \times (6 \times 2)$

Step 4: Simplify the fraction and addition

Now, calculate $\frac{20}{5}$:

$\frac{20}{5} = 4$

The expression becomes: $7 - \left\{ 5 + 10 \left[ 4 - 2 + 44 \right] - 8 \times 3^2 \right\} + 50 \times (6 \times 2)$

Now simplify inside the brackets: $4 - 2 + 44 = 46$

Thus, we have: $7 - \left\{ 5 + 10 \times 46 - 8 \times 3^2 \right\} + 50 \times (6 \times 2)$

Step 5: Simplify multiplications

Now perform the multiplications:

• $10 \times 46 = 460$
• $3^2 = 9$, so $8 \times 9 = 72$
• $50 \times (6 \times 2) = 50 \times 12 = 600$

So now the expression is: $7 - \left\{ 5 + 460 - 72 \right\} + 600$

Step 6: Simplify inside the braces

Now simplify inside the braces: $5 + 460 - 72 = 393$

So the expression becomes: $7 - 393 + 600$

Step 7: Final simplification

Now do the addition and subtraction: $7 - 393 = -386$ $-386 + 600 = 214$

Final Answer:

The value of the expression is 214.

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Would you like more details on how to approach such problems, or is there another step you'd like to go deeper into?

Related Questions:

1. How do you simplify nested parentheses in complex expressions?
2. What is the order of operations, and why is it important in expressions like this?
3. Can you explain how to evaluate powers, like $3^2$, in these kinds of problems?
4. How would this expression change if the signs inside the parentheses were reversed?
5. How do you handle fractions and multiplication in a multi-step problem?

Tip: Always handle operations inside parentheses first, followed by exponents, multiplication, and division (left to right), then addition and subtraction (left to right).