Question

Evaluate the expression. $$ [(12 \cdot 9) \div 6]-13.5 $$

Short answer

The evaluated expression is 4.5

Step-by-step solution

Step 1. Start inside the brackets

First, complete the multiplication operation inside the brackets. So \(12 \cdot 9 = 108\). Then, the expression will be \((108 \div 6) - 13.5\).

Step 2. Continue with the division

Next, perform the division operation. \(108 \div 6 = 18\). Now, the expression simplifies to \(18 - 13.5\).

Step 3. Perform the subtraction

Lastly, subtract the 13.5 from the 18. Hence, \(18 - 13.5 = 4.5\).

Key concepts

Order of Operations

Understanding the order of operations is crucial when evaluating mathematical expressions. It's the set of rules that tell us the sequence in which we should solve a math problem containing multiple steps.

The most common acronym to remember this sequence is PEMDAS, which stands for Parentheses, Exponents (or Orders in some countries), Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). In our exercise, we first deal with the operation within the brackets \(12 \cdot 9 = 108\), followed by the division \(108 \div 6 = 18\), and finally, we perform the subtraction to get our answer \(18 - 13.5 = 4.5\).

It's essential to follow these steps in the correct order; otherwise, we could end up with a different, incorrect result. For instance, if one were to subtract before dividing, they would not arrive at the correct answer for the given expression.

Arithmetic Operations

The foundation of most mathematical problems involves arithmetic operations. These include addition, subtraction, multiplication, and division. Each operation has its own set of rules and methods for carrying out the solution.

In our example, we encounter multiplication \(12 \cdot 9\) and division \(108 \div 6\), followed by subtraction \(18 - 13.5\). Multiplication involves combining equal groups, division is the process of distributing a number into equal parts, and subtraction is taking away one amount from another.

Common Mistakes

It's important to carry out multiplication and division with care, as they can involve large numbers or fractions, which can be tricky. Similarly, when subtracting numbers, especially with decimals, aligning the decimal points ensures accuracy.

Simplifying Expressions

When we talk about simplifying expressions, we refer to the process of performing operations to rewrite an expression in its most basic form. It means combining like terms, using properties of arithmetic to make expressions easier to interpret, or eliminating unnecessary complexity.

Our initial problem \( [(12 \cdot 9) \div 6]-13.5 \) becomes much simpler when we follow the order of operations. After multiplication and division, the expression is reduced to a single subtraction of a whole number from a decimal \(18 - 13.5\), giving us a straightforward result of 4.5.

The goal of simplification is not to change the value of the expression but to make it as clear and concise as possible. This is particularly important when dealing with more complex algebraic expressions where clarity can influence your ability to solve equations or understand the relationships between variables.