在横线上填写每一步的运算依据：
$\begin{aligned}
& 22+(-8)+(-2)+4 \\
= & 22+4+(-8)+(-2) \\
= & (22+4)+[(-8)+(-2)] .
= & 26+(-10) \\
= & 16 .
\end{aligned}$
Answer (2)
The problem uses the commutative property to rearrange the terms and the associative property to group them. By performing the addition step-by-step, we find that the final result of the expression is 16. This solution demonstrates the effectiveness of using properties of addition in mathematical calculations.
;
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 completes the calculation: 26 + ( − 10 ) = 16 .
Explanation
Analyzing the Problem We are given a series of arithmetic operations and asked to identify the property or rule used in each step. Let's analyze each step carefully.
Applying Commutative Property Step 1: 22 + ( − 8 ) + ( − 2 ) + 4 = 22 + 4 + ( − 8 ) + ( − 2 ) . Here, the order of the numbers is changed. This is allowed by the commutative property of addition, which states that a + b = b + a for any numbers a and b .
Applying Associative Property Step 2: 22 + 4 + ( − 8 ) + ( − 2 ) = ( 22 + 4 ) + [( − 8 ) + ( − 2 )] . In this step, the numbers are grouped using parentheses. This is allowed by the associative property of addition, which states that ( a + b ) + c = a + ( b + c ) for any numbers a , b , and c .
Performing Addition Step 3: ( 22 + 4 ) + [( − 8 ) + ( − 2 )] = 26 + ( − 10 ) . Here, the addition within the parentheses is performed: 22 + 4 = 26 and ( − 8 ) + ( − 2 ) = − 10 .
Final Calculation Step 4: 26 + ( − 10 ) = 16 . Finally, the last addition is performed: 26 + ( − 10 ) = 16 .
Examples
The commutative and associative properties are fundamental in various real-life scenarios. For instance, when calculating the total cost of items in a shopping cart, the order in which you add the prices doesn't matter (commutative property). Similarly, when grouping expenses for budgeting, whether you group rent and utilities first or groceries and transportation, the total expense remains the same (associative property). These properties simplify calculations and ensure accuracy in everyday tasks.
