Problem 77 - Algebra for College Students

Question

Simplify each numerical expression. $$-3[5-(-2)]-2(-4-9)$$

Step-by-Step Solution

Answer

The simplified expression is 5.

Step 1: Simplify Inside Brackets

Start by simplifying the expression inside the brackets: \(5 - (-2)\). When subtracting a negative number, it becomes addition. Thus, \(5 - (-2) = 5 + 2 = 7\).

Step 2: Evaluate the Expression with Simplified Brackets

Replace the simplified expression back into the main expression: \[-3[7] - 2(-4 - 9)\] This simplifies to \[-3 imes 7 - 2(-4 - 9)\].

Step 3: Simplify the Second Parentheses

Next, simplify the expression inside the second parentheses: \(-4 - 9\) becomes \(-13\). Thus, the expression is now \[-3 imes 7 - 2(-13)\].

Step 4: Perform Multiplications

Calculate the multiplications in the expression:1. \(-3 imes 7 = -21\)2. \(-2 imes (-13) = 26\)Substituting these results into the expression:\[-21 + 26\].

Step 5: Final Calculation

Combine the results of the multiplications:\(-21 + 26 = 5\).

Key Concepts

Brackets

Brackets in mathematical expressions help group numbers and operations together to clarify the order in which calculations should be performed. They are an essential part of managing complex expressions. When dealing with brackets, always start by simplifying the expression inside them first. This is because calculations within brackets have a higher priority than those outside them.

In our example, the expression inside the brackets was \(5 - (-2)\). It's important to look for any operations within the brackets and perform them before moving on. Here, subtracting a negative number changes the operation to addition, simplifying to \(7\).

Simplifying expressions inside brackets first avoids errors, as it ensures that you are performing operations in the correct sequence. After you've simplified what's inside the brackets, you can replace that simplified result back into the main expression.

Negative Numbers

Working with negative numbers in mathematics can sometimes be confusing, but they are crucial for a variety of calculations. Negative numbers have their own set of rules, especially when it comes to adding, subtracting, multiplying, and dividing.

One key rule is that subtracting a negative number is the same as adding the absolute value of that number. For instance, in our problem, the expression \(-2\) being subtracted involved converting it to an addition: \(5 - (-2) = 5 + 2\).

When multiplying negative numbers, a negative times a negative gives a positive result, while a negative times a positive gives a negative result. Thus, in the expression \(-2 \times (-13)\), the multiplication yields a positive \(26\).

Understanding these rules helps in simplifying expressions effectively and accurately, especially when negatives are combined with other operations like multiplication or division.

Order of Operations

Order of operations is critical when simplifying mathematical expressions. It ensures that everyone solves a problem consistently and gets the same result. The standard order of operations can be remembered using the acronym PEMDAS: Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).

In the provided exercise, the order guided us through simplifying the expression correctly. First, we dealt with the expressions inside the brackets (parentheses). Then, we handled any multiplications, and finally, performed the addition and subtraction.

• First, simplify inside the brackets: \([-3[7] - 2(-4 - 9)]\)
• Next, perform any multiplications: \(-3\times 7\) and \(-2\times (-13)\)
• Lastly, add or subtract the results as necessary: \(-21 + 26\)

This process avoids errors and maintains the logical flow of operations. Remembering the principle of PEMDAS helps not just with simplifying expressions, but also in understanding and solving complex mathematical problems.