Problem 7 - Elementary Algebra

Question

For the following problems, simplify the expressions. $$ (4+5)(4+6)-(4+7) $$

Step-by-Step Solution

Answer

Question: Simplify the expression \(((4+5)(4+6))-(4+7)\) using the order of operations (PEMDAS/BODMAS). Answer: 79

Step 1: Identify operations in parentheses

First, we'll identify the operations inside parentheses that need to be performed before the other operations. In this case, we have three operations inside parentheses: \((4+5)\), \((4+6)\) and \((4+7)\). We'll evaluate these operations first.

Step 2: Evaluate the operations in parentheses

Next, we'll perform the operations inside parentheses: \((4+5) = 9\), \((4+6) = 10\), and \((4+7) = 11\). Now, our expression looks like this: $$ (9)(10) - (11) $$

Step 3: Perform multiplication

According to the order of operations, we'll now perform the multiplication operation: $$ (9)(10) = 90 $$ Our expression now looks like this: $$ 90 - 11 $$

Step 4: Perform subtraction

Finally, we'll perform the subtraction operation: $$ 90 - 11 = 79 $$ The simplified expression is 79.

Key Concepts

Order of Operations

When simplifying mathematical expressions, it's essential to follow the order of operations to avoid any mistakes. This is often remembered by the acronym PEMDAS:

• P: Parentheses first
• E: Exponents (ie. powers and roots, etc.)
• M/D: Multiplication and Division (left-to-right)
• A/S: Addition and Subtraction (left-to-right)

In any expression, we first handle the operations within parentheses, then move on to exponents. After these are resolved, we tackle multiplication and division, followed by addition and subtraction as they appear from left to right. By following this order, we ensure that each operation is performed correctly, maintaining the mathematical integrity of the expression.

In our problem, the expression \((4+5)(4+6)-(4+7)\) relies on this fundamental rule to ensure that calculations are done properly.

Parentheses Evaluation

The use of parentheses signifies that the operations inside them should be performed first. This is the initial step in the order of operations. Parentheses can contain any combination of numbers and operations that need to be resolved before moving forward. In the expression given, parentheses are applied around each of the sums:

• \((4+5)\)
• \((4+6)\)
• \((4+7)\)

By solving what's inside the parentheses first, we simplify the expression to manageable numbers. For example, \(4+5\) becomes \(9\), \(4+6\) becomes \(10\), and \(4+7\) turns into \(11\). These simplified results then act as the new values to be used in subsequent operations.

Multiplication and Subtraction

After simplifying the expression by evaluating the parentheses, the next operations to consider are multiplication and subtraction. According to the order of operations (PEMDAS), multiplication comes before subtraction.

In the steps outlined, we first multiply the results from the evaluated parentheses: \(9\times10\), which gives \(90\). This multiplication simplifies the expression further, leaving us with \(90-11\).

Finally, we perform the subtraction. This involves subtracting \(11\) from \(90\), resulting in \(79\).

Each of these operations follows logically from the previous one, reducing the expression to a simple final number. Understanding the precise order ensures no steps are skipped and every operation is correctly executed.