Problem 52 - 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. $$-5(-2-8)-4(6-10)$$

Step-by-Step Solution

Answer

The expression simplifies to 66.

Step 1: Solve Parentheses

First, solve the expressions inside the parentheses. We have two sets of parentheses:i) -2 - 8 = -10ii) 6 - 10 = -4Now the expression becomes: \[-5(-10) - 4(-4)\]

Step 2: Multiplication

Next, perform the multiplication for each term:i) -5(-10) = 50Multiply -5 with -10, the negative signs cancel to give a positive product.ii) -4(-4) = 16Multiply -4 with -4, again the negative signs cancel.After performing these multiplications, the expression becomes:\[50 + 16\]

Step 3: Addition

Finally, perform the addition of the results from the multiplications: Add 50 and 16 together:\[50 + 16 = 66\]

Key Concepts

Simplifying Expressions

Simplifying expressions is all about making complex mathematical phrases easier to solve or understand. When you first look at an expression like \(-5(-2-8)-4(6-10)\), it can seem a bit intimidating. But by focusing on one part of the equation at a time, it becomes more manageable.

First, tackle what's inside the parentheses. This limits the scope of the problem and ensures you're tackling sections you can solve readily. For instance:

• -2 minus 8 simplifies to -10.
• Similarly, 6 minus 10 simplifies to -4.

Once the parentheses are simplified, the expression changes, often making the next steps clearer.

Remember that simplifying isn't just about doing calculations; it's also about organizing the expression logically. Breaking down and solving smaller parts leads to an easier overall problem.

Negative Numbers

Negative numbers can be confusing because they change the rules a bit when compared to positive numbers. It's important to remember a few key points when dealing with them:

• Two negatives make a positive: When you multiply or divide two negative numbers, the negatives cancel out, resulting in a positive number. For instance, -5 multiplied by -10 is 50.
• When adding or subtracting negative numbers: Adding a negative is quite similar to subtracting. For example, 5 + (-3) is the same as 5 - 3.

Understanding these properties helps in simplifying expressions accurately, ensuring you're handling each step correctly.

Multiplication and Addition Rules

Multiplication and addition are fundamental operations in mathematics and are crucial when simplifying expressions. Here are some rules to keep in mind:

• Always handle multiplication before addition due to the order of operations: In any mathematical expression, multiplication and division should be completed before addition and subtraction, unless there are parentheses altering the sequence.
• Order doesn’t matter for addition: Unlike subtraction, addition can be done in any order. So, \(50 + 16\) is the same as \(16 + 50\).
• Check signs with multiplication: Always pay attention to the signs of the numbers being multiplied. If both numbers are negative, their product will be positive.

By sticking to these rules, you ensure that the process of simplifying expressions remains consistent and accurate, leading to the correct solution.