Problem 22 - Intermediate Algebra

Question

Simplify each numerical expression. Be sure to take advantage of the properties whenever they can be used to make the computations easier. $$ -86[49+(-48)] $$

Step-by-Step Solution

Answer

The simplified expression is \(-86\).

Step 1: Simplify the Expression Inside the Brackets

First, look at the expression inside the brackets: \(49 + (-48)\). This is a simple addition/subtraction problem. You can rewrite this as \(49 - 48\). Calculate the result: \(49 - 48 = 1\). So, the expression inside the brackets simplifies to 1.

Step 2: Apply the Simplification to the Entire Expression

Now that the expression inside the brackets is simplified to 1, substitute it back into the original expression: \(-86[49 + (-48)]\) becomes \(-86[1]\).

Step 3: Multiply the Simplified Terms

Multiply \(-86\) by the simplified expression in the brackets, which is 1. \(-86 \times 1 = -86\).

Step 4: Conclusion

The original expression \(-86[49 + (-48)]\) simplifies to \(-86\).

Key Concepts

Properties of Operations

The properties of operations in mathematics help simplify complex expressions and make calculations easier. Understanding these properties is essential when working with expressions. Let’s break down some of the key properties:

• Commutative Property: This property applies to addition and multiplication. It states that the order in which you add or multiply numbers does not affect the result. For example, \(a + b = b + a\) and \(a \times b = b \times a\).
• Associative Property: This property allows us to group numbers in a different way when adding or multiplying without changing the result. For example, \((a + b) + c = a + (b + c)\) and \((a \times b) \times c = a \times (b \times c)\).
• Distributive Property: It is used to multiply a number by a sum. For example, \(a(b + c) = ab + ac\). This property is especially useful when simplifying expressions with brackets.

Recognizing and applying these properties means you can easily restructure and simplify expressions without altering their meaning or result.

Addition and Subtraction

Addition and subtraction are fundamental operations in mathematics, and often they are the first steps in simplifying expressions within brackets. In the given problem, we deal with addition and subtraction inside the brackets:

• The expression \(49 + (-48)\) is essentially a subtraction problem. Since adding a negative number is the same as subtracting, it can be rewritten as 49 - 48.
• Simplifying such operations within the brackets makes it easier to tackle the rest of the expression. Here, 49 - 48 = 1.

This simplification helps in reducing the complexity of the original expression.

Multiplication

Multiplication is one of the basic operations in arithmetic and is frequently encountered in simplifying expressions. Here, multiplication takes place after simplifying the expression inside the brackets. Once we've determined that \(49 + (-48) = 1\), we substitute 1 back into the main expression:

• Now the expression becomes \(-86[1]\), meaning we need to multiply \(-86\) by the result from the brackets.
• When you multiply \(-86\) by 1, you utilize the property that any number multiplied by 1 remains unchanged. Thus, \(-86 \times 1 = -86\).

By applying the multiplication step, we complete the simplification of the original expression. So, we see that the expression \(-86[49 + (-48)]\) is simplified to \(-86\).