―5( ―5( ―2(5 ― 3) ― (3 ― 5)) + 5) ― 5(5 ― 2)

Let's simplify the given expression step by step:

Given expression:

\[
-5 \left( -5 \left( -2(5 - 3) - (3 - 5) \right) + 5 \right) - 5(5 - 2)
\]

### Step 1: Simplify the innermost parentheses
Start by simplifying the expressions inside the innermost parentheses:

\[
5 - 3 = 2
\]
\[
3 - 5 = -2
\]

Now, substitute these values back:

\[
-5 \left( -5 \left( -2(2) - (-2) \right) + 5 \right) - 5(5 - 2)
\]

### Step 2: Perform the multiplications
Next, calculate \( -2 \times 2 \):

\[
-2 \times 2 = -4
\]

So the expression becomes:

\[
-5 \left( -5 \left( -4 + 2 \right) + 5 \right) - 5(5 - 2)
\]

### Step 3: Simplify inside the parentheses
Now, simplify \( -4 + 2 \):

\[
-4 + 2 = -2
\]

Substitute back:

\[
-5 \left( -5(-2) + 5 \right) - 5(5 - 2)
\]

### Step 4: Continue with multiplications
Now, calculate \( -5 \times -2 \):

\[
-5 \times -2 = 10
\]

So the expression now is:

\[
-5 \left( 10 + 5 \right) - 5(5 - 2)
\]

Simplify the addition:

\[
10 + 5 = 15
\]

So:

\[
-5(15) - 5(5 - 2)
\]

### Step 5: Simplify the remaining expressions
Calculate the value of \( 5 - 2 \):

\[
5 - 2 = 3
\]

So the expression is:

\[
-5(15) - 5(3)
\]

### Step 6: Final multiplication
Now, perform the multiplications:

\[
-5 \times 15 = -75
\]
\[
-5 \times 3 = -15
\]

So the expression becomes:

\[
-75 - 15
\]

### Step 7: Final addition
Now, add the two negative numbers:

\[
-75 - 15 = -90
\]

### Final Answer:

\[
\boxed{-90}
\]

Would you like more details on any of these steps?

Here are five related questions:
1. How do you handle nested multiplications in algebraic expressions?
2. What steps are involved in simplifying an expression with multiple terms and operations?
3. How does the distributive property apply in complex expressions?
4. Can you explain why subtracting a negative number results in addition?
5. What are some common mistakes to avoid when simplifying expressions with multiple operations?

**Tip:** Always keep track of signs carefully when simplifying expressions, especially when dealing with nested operations.

Learn how to simplify the algebraic expression -5( -5( -2(5 - 3) - (3 - 5)) + 5) - 5(5 - 2) step-by-step. This problem involves algebraic simplification with nested parentheses and integer multiplication. Suitable for students in Grades 7-9.

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