Problem 37 - Prealgebra

Question

Use the rule for order of operations along with the rules for addition, subtraction, and multiplication to simplify each of the following expressions. $$-7+2[-5-9]$$

Step-by-Step Solution

Answer

The simplified expression is -35.

Step 1: Simplify Inside the Brackets

First, address the operation inside the brackets \[-5 - 9\].Calculate \[-5 - 9 = -14\]. This simplifies the expression to \[-7 + 2(-14)\].

Step 2: Apply Multiplication

Next, handle the multiplication part of the expression \[2(-14)\].Multiply 2 by -14, which gives \[2 imes -14 = -28\]. Now, the expression is simplified to \[-7 - 28\].

Step 3: Perform Addition/Subtraction

Finally, perform the subtraction \[-7 - 28\].Subtracting 28 from -7 gives \[-35\].

Key Concepts

Addition

Addition is one of the foundational operations in mathematics. It involves combining two or more numbers to get a total sum. When you're performing arithmetic operations, it helps to think of addition as moving numbers forward on a number line. For example, adding 3 to 5, as in the expression \(5 + 3\), means you start at 5 and move 3 steps forward to 8.

• Addition is remarkable for being commutative: \(a + b = b + a\).
• It is also associative, meaning \((a + b) + c = a + (b + c)\).
• The identity element for addition is 0, so \(a + 0 = a\).

In complex expressions, addition often comes after parentheses, exponents, multiplication, and division in the order of operations. Always make sure calculations inside brackets or parentheses are concluded before performing addition.

Subtraction

Subtraction, the inverse operation of addition, is used to find the difference between numbers. It is as simple as thinking about taking away or moving backward on a number line. For instance, in the operation \(9 - 5\), you start at 9 and move 5 steps backward, ending up at 4.

• Unlike addition, subtraction is not commutative, which means \(a - b eq b - a\).
• Subtraction is also not associative, so \((a - b) - c eq a - (b - c)\).
• The role of subtraction often demands attention, especially in sequences or series of operations.

In our exercise, correctly handling subtraction within brackets and later in the simplified form is vital. Always manage carefully where negative signs might alter the outcome, especially in complex expressions.

Multiplication

Multiplication is the process of adding a number to itself a particular number of times. In simple terms, it's repeated addition. For example, \(4 \times 3\) is the same as adding 4 three times: \(4 + 4 + 4\). Multiplication is an essential operation in arithmetic and is often used when calculating area and volume.

• One key property of multiplication is that it is commutative: \(a \times b = b \times a\).
• It is also associative: \((a \times b) \times c = a \times (b \times c)\).
• Multiplication has a distributive property over addition: \(a \times (b + c) = a \times b + a \times c\).

In terms of order of operations, always perform multiplication before addition and subtraction unless directed by brackets. This ensures accurate simplifications and results in expressions.