Let's break down the expression step by step:

Given expression:

\[
(22 - 42^2 + 62 - 8) + (-r + 52 - 3z + 1) \times 2
\]

### Step 1: Simplify the expression inside the first set of parentheses.

\[
22 - 42^2 + 62 - 8
\]

Calculate \(42^2\):

\[
42^2 = 1764
\]

Now, substitute and simplify:

\[
22 - 1764 + 62 - 8 = (22 + 62 - 8) - 1764
\]

\[
22 + 62 = 84
\]

\[
84 - 8 = 76
\]

\[
76 - 1764 = -1688
\]

So, the first part simplifies to \(-1688\).

### Step 2: Simplify the expression inside the second set of parentheses and multiply by 2.

\[
-r + 52 - 3z + 1
\]

Simplify:

\[
52 + 1 = 53
\]

So,

\[
-r + 53 - 3z
\]

Now multiply by 2:

\[
2(-r + 53 - 3z) = -2r + 106 - 6z
\]

### Step 3: Combine the two parts.

\[
-1688 + (-2r + 106 - 6z)
\]

Simplify:

\[
-1688 + 106 - 2r - 6z
\]

Final simplified expression:

\[
-1582 - 2r - 6z
\]

This is the final simplified form of the given expression.

Would you like any further details or explanations?

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Here are 5 related questions to consider:

1. How would the expression change if \(r = 0\) and \(z = 0\)?
2. What is the effect on the expression if \(r\) and \(z\) are positive integers?
3. How can you verify the result using the distributive property?
4. What happens if the coefficients of \(r\) and \(z\) are increased?
5. How would you solve this expression if it was an equation set to zero?

**Tip:** When simplifying expressions, always perform operations within parentheses first and carefully follow the order of operations (PEMDAS: Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).

Learn how to simplify the expression (22 - 42^2 + 62 - 8) + (-r + 52 - 3z + 1) * 2 step-by-step with detailed explanations. This problem involves algebraic operations, including simplifying expressions with parentheses and integer operations. Suitable for high school students.

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