Problem 43 - 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. $$(8-3)(2-7)$$

Step-by-Step Solution

Answer

The expression simplifies to \(-25\).

Step 1: Evaluate Inside the Parentheses

Start by simplifying the expressions inside the parentheses. For the first set of parentheses, calculate \(8 - 3\). For the second set of parentheses, calculate \(2 - 7\). So, we get: \(8 - 3 = 5\) and \(2 - 7 = -5\). The expression becomes \((5)(-5)\).

Step 2: Perform Multiplication

Now, multiply the results of the numbers obtained from inside the parentheses. That is, calculate \(5 \times (-5)\). The result of this multiplication is \(-25\).

Key Concepts

Addition and Subtraction

Understanding addition and subtraction is essential when dealing with algebraic expressions. These operations allow us to combine numbers and find differences, which is often the first step in tackling complex problems.

Addition simply involves combining two or more numbers to arrive at a total. For example, when we add 2 and 3, the result is 5. It’s like collecting apples in a basket; each apple added increases the total count.

Subtraction, on the other hand, is all about finding the difference between numbers. When you subtract, you are essentially taking small bites out of a whole. For example, starting with 8 and subtracting 3 leaves you with 5. Imagine it as removing those apples one at a time from your basket.

When simplifying expressions, always remember to prioritize operations inside the parentheses. The exercise provided demonstrates this beautifully by first simplifying the operations inside the parentheses from the expression \((8-3)(2-7)\).

You calculate:\[\begin{align*}8 - 3 &= 5\2 - 7 &= -5\end{align*}\]
This simplification sets up the expression for the next step: multiplication. Make sure to carry out each subtraction fully and accurately, as errors here can lead to mistakes in the final result.

Multiplication

After simplifying expressions inside parentheses comes multiplication. Multiplication combines numbers similarly to repeated addition, but more efficiently.

Consider the expression \((5)(-5)\) from our exercise. Here, you multiply 5 by -5.

It's crucial to remember the rules around multiplying positive and negative numbers:

• If you multiply a positive number by a positive number, the result is positive (e.g., 3 × 4 = 12).
• If you multiply a positive number by a negative number, the result is negative (e.g., 3 × -4 = -12).
• If you multiply a negative number by a positive number, the result is also negative (e.g., -3 × 4 = -12).
• Finally, if you multiply two negative numbers, the result is positive (e.g., -3 × -4 = 12).

Using these rules, calculate \(5 \times (-5)\), which equals -25.

Multiplication can feel like a powerful tool, allowing us to scale and transform numbers in our expressions. Mastery of this operation helps handle more complex equations and expressions efficiently.

Simplifying Expressions

Simplifying expressions is one of the foundational skills in algebra, allowing us to break down complex problems into manageable parts. It involves applying the order of operations and making expressions as concise as possible.

The order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)), helps determine the sequence in which to perform operations:

• Start with calculations inside Parentheses.
• Proceed with Exponents (not applicable in this problem).
• Follow with Multiplication and Division.
• Finally, complete Addition and Subtraction.

In the provided exercise \((8-3)(2-7)\), we started by simplifying inside the parentheses, obtaining \(5\) and \(-5\).

Next, we moved to multiplication, calculating \(5 \times (-5)\), resulting in \(-25\). Through simplification, this problem illustrates how efficiently breaking down expressions with the correct order of operations can lead to the correct solution.

Simplifying expressions forms a strong base for more advanced algebraic concepts and ensures clarity in mathematical problem-solving.