$$  $$$$ \left\lbrack\left(+5\right)^3\div\left(-5\right)\cdot2+7\cdot\left(13-2\cdot3\right)\right\rbrack\cdot\left\lbrace\left\lbrack\left(-4^{}\right)^3-8^2\cdot\left(-4\right)+5.24\right\rbrack\div10^2\right\rbrace $$$$  $$

Let's break this expression down step by step:

\[
\left[ \left( +5 \right)^3 \div \left( -5 \right) \cdot 2 + 7 \cdot \left( 13 - 2 \cdot 3 \right) \right] \cdot \left\{ \left[ \left( -4 \right)^3 - 8^2 \cdot \left( -4 \right) + 5.24 \right] \div 10^2 \right\}
\]

### Step 1: Solve the first bracket $\left[ \cdot \right]$
\[
\left( +5 \right)^3 = 5^3 = 125
\]
\[
125 \div (-5) = -25
\]
\[
-25 \cdot 2 = -50
\]
Next, calculate $ 7 \cdot \left( 13 - 2 \cdot 3 \right) $:
\[
2 \cdot 3 = 6
\]
\[
13 - 6 = 7
\]
\[
7 \cdot 7 = 49
\]
Now sum everything in the first bracket:
\[
-50 + 49 = -1
\]

### Step 2: Solve the second bracket $\left\{ \cdot \right\}$
First, solve $\left[ \left( -4 \right)^3 - 8^2 \cdot \left( -4 \right) + 5.24 \right]$:
\[
\left( -4 \right)^3 = -64
\]
\[
8^2 = 64
\]
\[
64 \cdot (-4) = -256
\]
Now sum up inside the second bracket:
\[
-64 - (-256) + 5.24 = -64 + 256 + 5.24 = 197.24
\]
Now divide by $ 10^2 = 100 $:
\[
\frac{197.24}{100} = 1.9724
\]

### Step 3: Multiply the results of both brackets
\[
(-1) \cdot 1.9724 = -1.9724
\]

Thus, the value of the expression is:

\[
\boxed{-1.9724}
\]

Would you like any further details or explanations?

Here are 5 related questions to explore:
1. How does the order of operations affect complex expressions like this?
2. What are some common mistakes when dealing with negative numbers in exponents?
3. How do powers of 10 affect division in large expressions?
4. What role does distribution play in simplifying nested brackets?
5. How can we simplify expressions involving both multiplication and division with negative numbers?

**Tip**: Always handle powers and parentheses first to avoid calculation errors when evaluating complex expressions.

$$ \left[\left(+5\right)^3\div\left(-5\right)\cdot2+7\cdot\left(13-2\cdot3\right)\right]\cdot\left\{\left[\left(-4^{}\right)^3-8^2\cdot\left(-4\right)+5.24\right]\div10^2\right\} $$

This problem involves solving a complex mathematical expression with nested brackets, exponents, and various operations. Learn step-by-step how to handle operations like powers, division, multiplication, and addition, using the order of operations (PEMDAS). Suitable for high school students in Grades 9-12.

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