Problem 24 - Beginning and Intermediate Algebra

Question

Simplify each expression. \(5(6-2)\)

Step-by-Step Solution

Answer

The simplified expression is 20.

Step 1: Understand the Expression

The given expression is \(5(6-2)\). It consists of a multiplication between a number 5 and the result of the subtraction inside the parentheses \((6-2)\). Our goal is to simplify this expression by first simplifying the operation inside the parentheses.

Step 2: Simplify Inside the Parentheses

The first operation to perform is inside the parentheses. Calculate \(6-2\) which equals 4. So, the expression is transformed into \(5 \times 4\).

Step 3: Perform the Multiplication

Now, multiply the result from the previous step by 5: \(5 \times 4 = 20\).

Step 4: Solution Complete

The simplified expression is \(20\).

Key Concepts

Order of Operations

When simplifying expressions, it's important to follow the **order of operations**. This rule tells us the sequence to perform operations to arrive at the correct solution. Memorize the acronym PEMDAS or BODMAS to help remember:

• **P**arentheses or **B**rackets
• **E**xponents or **O**rders (such as squares or square roots)
• **M**ultiplication and **D**ivision (left to right)
• **A**ddition and **S**ubtraction (left to right)

So, in the expression \(5(6-2)\), we first deal with the operation inside the parentheses because parentheses come first in PEMDAS. By following the order of operations, we ensure that the calculations are done in the correct order and the right result is achieved. Ignoring these rules can lead to wrong answers. Hence, always give priority to parentheses and then follow down the list of operations.

Parentheses in Math

Using **parentheses** in mathematics is a way of showing which operations should be done first. In our example, \(5(6-2)\), the parentheses tell us that we should perform the subtraction \((6-2)\) before multiplying by 5.Why are parentheses crucial?

• They clarify the order of operations.
• They allow for grouping, which changes how calculations are performed.
• They prevent misunderstandings in complex expressions.

When you see an expression with parentheses, always handle them first, simplifying inside to make your expression easier to solve. Once the operation inside the parentheses is complete, the expression can be condensed into a simpler form, like turning \(5(6-2)\) into \(5 \times 4\). This makes it easy for the subsequent arithmetic operations.

Basic Arithmetic Operations

**Basic arithmetic operations** include addition, subtraction, multiplication, and division. These are the foundational operations we continually use in mathematics. In the expression \(5(6-2)\), two of these basic operations are used: subtraction and multiplication.Here's how they work in this context:

• **Subtraction**: Inside the parentheses, \(6-2\) simplifies to 4. Subtraction involves taking one number away from another, reducing it to a smaller number.
• **Multiplication**: After simplifying the parentheses, you multiply 5 by 4. In multiplication, numbers are repeated added. So \(5 \times 4\) means you add 4 together 5 times, which equals 20.

Understanding each of these operations and knowing their proper order ensures that expressions are simplified correctly. Basic operations are integral in solving any mathematical problem, so mastering them is key to solving more complex equations down the line.