Problem 59 - Algebra for College Students

Question

Simplify each of the numerical expressions. $$(17-12)(13-9)(7-4)$$

Step-by-Step Solution

Answer

The simplified expression is 60.

Step 1: Simplify Each Subtraction Expression

Begin by simplifying each of the subtraction operations in the expression \((17-12)(13-9)(7-4)\). Calculate:- \(17-12 = 5\)- \(13-9 = 4\)- \(7-4 = 3\)

Step 2: Multiply Simplified Numbers Together

Now multiply the simplified results together: - First, multiply the first two results: \(5 \times 4 = 20\)- Next, multiply the result with the last number: \(20 \times 3 = 60\)

Key Concepts

Simplifying Expressions

Simplifying expressions is like cleaning your room – you're organizing and making it easier to manage. In mathematics, simplification means making an expression easier to understand or solve. For numerical expressions, simplifying often involves performing arithmetic operations like addition, subtraction, multiplication, or division to reduce the expression to a single value.

In our example, \[(17-12)(13-9)(7-4)\]the expression contains several operations that need to be simplified.

Here’s what to do:

• First, focus on breaking down parts of the expression that can be calculated. This will involve tackling each subtraction individually.
• Once you have simplified each section, you can then combine the simplified parts through operations like multiplication.

By following these steps, we transform the original complex expression into something much more straightforward.

Subtraction Operations

Subtraction is the process of finding the difference between numbers. It is a fundamental arithmetic operation. Subtraction helps simplify expressions by reducing terms to a more manageable state. In algebra and numeric expressions, paying close attention to subtraction helps avoid mistakes.

In the expression \[(17-12)(13-9)(7-4),\]we handle subtraction first to lessen complexity:

• Start from left to right. Simplify each subtraction inside the parentheses.
• Calculate each step carefully. For instance, \(17-12\) results in \(5\).
• Continue with the next operations — \(13-9 = 4\) and \(7-4 = 3\).

By completing the subtraction first, you transform the expression into something you can work with easier. Once subtraction is completed, you move on to the next step: multiplication.

Multiplication

Multiplication is a process of adding a number to itself a certain number of times. It is represented by the symbol \(\times\) and sometimes by simple juxtaposition.Multiplication is essential for simplifying expressions, especially when combined with other arithmetic operations.

After simplifying by subtraction, we used multiplication in the exercise:\[(17-12)(13-9)(7-4).\]We already found the simplified numbers: \(5\), \(4\), and \(3\). Now we multiply them:

• First, multiply the first two numbers: \(5\times4=20\).
• Then, take the result and multiply by the next number: \(20\times3=60\).

By carefully performing these multiplications, we arrive at the final simplified value of \(60\). Multiplication consolidates the expression, showcasing the combined value of all parts.