Problem 46 - 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. $$(3-7)(2-8)$$

Step-by-Step Solution

Answer

The simplified expression is 24.

Step 1: Simplify Inside the Parentheses

First, simplify the expressions within each set of parentheses. You have two expressions: 1) Inside the first parentheses: \(3-7\) Simplify it to \(-4\).2) Inside the second parentheses: \(2-8\) Simplify it to \(-6\).The expression becomes: \((-4)(-6)\).

Step 2: Multiply Simplified Values

Now, multiply the simplified values from inside the parentheses. This involves multiplying two negative numbers:- Multiply \(-4\) and \(-6\).When you multiply two negative numbers, the product is positive: - \((-4) \times (-6) = 24\).

Step 3: Conclusion

The result of the multiplication gives the simplified form of the original expression:- \(24\).

Key Concepts

Simplifying Expressions

In mathematics, simplifying expressions involves making them as straightforward as possible without changing their value. Let’s look at how this works:
To simplify an expression like \((3-7)(2-8)\),
the first step is to deal with the operations inside the parentheses. This means we perform the subtraction for each pair:

• The expression inside the first set of parentheses, \(3-7\), simplifies to \(-4\).
• Similarly, the expression inside the second set of parentheses, \(2-8\), becomes \(-6\).

Now, our expression is reduced to \((-4)(-6)\),a more straightforward form ready for multiplication.
Simplifying before proceeding with further operations makes calculations easier and reduces errors.

Negative Numbers

Negative numbers can sometimes be tricky, but with a clear understanding, you will find them very manageable. A negative number is simply a number less than zero, and it behaves in specific ways during arithmetic calculations.
For subtraction, as seen in our examples:

• When you subtract a larger number from a smaller one, the result is negative, such as \(3-7\), which becomes \(-4\).
• Similarly, \(2-8\) simplifies to \(-6\).

Understanding these relationships helps you handle negative results with confidence.
When you multiply two negative numbers, the result becomes positive. Think of it this way: each negative sign "flips" the direction, and flipping twice brings you back to the positive direction.

Multiplication Rules

The rules of multiplication are straightforward and apply universally to all numbers, including negative numbers. Let’s break them down:

• Multiplying two positive numbers results in a positive number.
• Multiplying two negative numbers, like \((-4) imes (-6)\), also results in a positive number because two negatives make a positive.
• Multiplying a positive number by a negative number, or vice versa, results in a negative number.

In the given expression, after simplifying the numbers inside the parentheses, we multiplied \(-4\) and \(-6\).
Because of the rule that two negative numbers multiplied together yield a positive product, we ended up with \(24\).
These rules are fundamental and critical for solving more complex mathematical problems.