Problem 72 - Intermediate Algebra

Question

Simplify each of the numerical expressions. $$ [27-(4 \cdot 2+5 \cdot 2)][(5 \cdot 6-4)-20] $$

Step-by-Step Solution

Answer

The simplified expression is 54.

Step 1: Simplify Inside the Parentheses

Begin by simplifying the expression inside the first set of square brackets: \(27 - (4 \cdot 2 + 5 \cdot 2)\). This includes two multiplications. Calculate each:\(4 \cdot 2 = 8\) and \(5 \cdot 2 = 10\). Add these results: \(8 + 10 = 18\). Subtract from 27: \(27 - 18 = 9\). The simplified expression inside the first brackets is \(9\).

Step 2: Simplify the Second Bracket Expression

Next, simplify the expression inside the second set of square brackets: \((5 \cdot 6 - 4) - 20\). Start with multiplication: \(5 \cdot 6 = 30\). Subtract 4: \(30 - 4 = 26\). Then subtract 20: \(26 - 20 = 6\). The simplified expression inside the second brackets is \(6\).

Step 3: Multiply the Two Results

Finally, multiply the results from the simplified expressions:\[9 \cdot 6 = 54\].

Key Concepts

Simplifying Expressions

Simplifying expressions involves reducing a numerical or algebraic expression to its simplest or most basic form. This process helps make complex expressions easier to understand and work with. Let's break down what it means to simplify expressions and how to do it effectively.
First, identify and perform any operations that can be completed within the expression. This might involve simplifying terms that are grouped together in parentheses or brackets. In expression problems like the one given, start by looking for arithmetic operations like addition, subtraction, multiplication, and division.

• Simplify multiplication and division first since they are performed before addition and subtraction.
• Use parentheses or brackets to clearly show which operations should be done together.

Simplifying isn't just about doing calculations correctly but also organizing them in a coherent way. Efficient simplification aids in visualizing mathematical problems and solutions more clearly.

Order of Operations

When dealing with numerical expressions, it's crucial to follow the correct order of operations to arrive at the right solution. The order of operations is a set of rules that dictates the sequence in which we should perform arithmetic operations. Remember the acronym PEMDAS as a helpful guideline:

• P: Parentheses first – Start by solving expressions inside parentheses or brackets.
• E: Exponents (i.e., powers and roots, etc.)
• M/D: Multiplication and Division – From left to right, perform these operations next.
• A/S: Addition and Subtraction – Lastly, from left to right, perform these operations.

In the original problem, multiply arithmetic tasks inside each bracket before moving onto other operations, as seen when expressions were simplified by performing multiplications first. This process ensures accuracy, avoiding errors that can arise from incorrect operation sequences.

Algebraic Concepts

Algebra involves working with symbols and variables to represent numbers in equations and expressions. While the given exercise specifically deals with numerical expressions, the core principles of algebraic manipulation still apply.
Understanding algebraic concepts can make tackling these expressions more intuitive. Here are some foundational concepts:

• Variables: These are symbols that represent unknown values. Though not used explicitly here, preparing for algebraic expressions with variables involves simplifying expressions similar to numerical expressions.
• Constants: Numbers on their own are constants, unaffected by other variables or operations. Simplifying involves finding like terms and combining them.

Practicing simplification and the order of operations on numerical expressions sets a strong foundation for handling algebraic expressions efficiently. This approach helps in progressing towards more complex algebraic problems with confidence.