在横线上填写每一步的运算依据：
$\begin{aligned}
& 22+(-8)+(-2)+4 \\
= & 22+4+(-8)+(-2) \\
= & (22+4)+[(-8)+(-2)] \\
= & 26+(-10) \\
= & 16
\end{aligned}$

In the given expression, we apply the commutative property to rearrange terms, the associative property to group them, and then perform the additions to simplify the expression to 16. Each step demonstrates fundamental properties of addition that help in simplifying complex expressions. Understanding these properties enhances problem-solving skills in mathematics.
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Answered by Anonymous | 2025-07-04

The first step uses the commutative property to rearrange the terms: 22 + ( − 8 ) + ( − 2 ) + 4 = 22 + 4 + ( − 8 ) + ( − 2 ) .
The second step uses the associative property to group the terms: 22 + 4 + ( − 8 ) + ( − 2 ) = ( 22 + 4 ) + [( − 8 ) + ( − 2 )] .
The third step performs the addition within the parentheses: ( 22 + 4 ) + [( − 8 ) + ( − 2 )] = 26 + ( − 10 ) .
The final step performs the addition: 26 + ( − 10 ) = 16 .

Explanation

Understanding the Problem We are given the expression 22 + ( − 8 ) + ( − 2 ) + 4 and a step-by-step simplification. Our goal is to identify the mathematical property used in each step.

Applying Commutative Property Step 1: 22 + ( − 8 ) + ( − 2 ) + 4 = 22 + 4 + ( − 8 ) + ( − 2 ) . The order of the terms has been changed. This is an application of the commutative property of addition, which states that a + b = b + a .

Applying Associative Property Step 2: 22 + 4 + ( − 8 ) + ( − 2 ) = ( 22 + 4 ) + [( − 8 ) + ( − 2 )] . Here, the terms are grouped using parentheses. This is an application of the associative property of addition, which states that ( a + b ) + c = a + ( b + c ) .

Performing Addition Step 3: ( 22 + 4 ) + [( − 8 ) + ( − 2 )] = 26 + ( − 10 ) . In this step, we perform the addition within the parentheses: 22 + 4 = 26 and ( − 8 ) + ( − 2 ) = − 10 .

Performing Subtraction Step 4: 26 + ( − 10 ) = 16 . Finally, we perform the addition: 26 + ( − 10 ) = 26 − 10 = 16 .


Examples
The commutative and associative properties are fundamental in various real-life scenarios. For example, when calculating the total cost of items in a shopping cart, you can add the prices in any order (commutative property). Similarly, when grouping expenses in a budget, you can group them in different ways without changing the total amount (associative property). These properties simplify calculations and provide flexibility in problem-solving.

Answered by GinnyAnswer | 2025-07-04