Problem 49 - Fundamentals of Mathematics

Question

Find each value. Check each result with a calculator. \((4+7) \cdot(8-3)\)

Step-by-Step Solution

Answer

The value is 55.

Step 1: Solve the Expression Inside the First Parenthesis

First, solve the expression inside the first parenthesis: \(4 + 7\). This addition equals \(11\).

Step 2: Solve the Expression Inside the Second Parenthesis

Next, solve the expression inside the second parenthesis: \(8 - 3\). This subtraction equals \(5\).

Step 3: Multiply the Results

Now, multiply the results from Step 1 and Step 2: \(11 \cdot 5\). This multiplication gives \(55\).

Step 4: Check the Result with a Calculator

Using a calculator, enter the entire expression \((4+7) \cdot (8-3)\) and confirm that the result is \(55\). This verifies the solution is correct.

Key Concepts

Understanding Parentheses in Math

Parentheses in arithmetic expressions play a crucial role in determining the order of operations. Without them, mathematical expressions can become ambiguous. When you see parentheses, the operations within them must be completed first.
For example, consider the expression \((4+7) \cdot (8-3)\). Here:

• The addition \(4 + 7\) is completed first, resulting in \(11\).
• The subtraction \(8 - 3\) is completed next, resulting in \(5\).

This means the operations inside the parentheses are prioritized before any other operations like multiplication. Parentheses ensure calculations are performed in the correct sequence. In more complex situations, multiple sets of parentheses might be present, requiring you to work from the innermost set to the outermost. Always remember: managing parentheses accurately can drastically change the result of an expression.

Exploring Arithmetic Expressions

Arithmetic expressions consist of numbers, operators like addition, subtraction, multiplication, and division, and sometimes variables. The key to solving these correctly is understanding the "order of operations".
In our expression \((4+7) \cdot (8-3)\), we see an operation inside each set of parentheses. Here's how it breaks down:

• First, evaluate what is inside the parentheses.
• After solving those, focus on any multiplication or division from left to right.

For the given expression:

• Calculate \(4 + 7\) to get \(11\).
• Calculate \(8 - 3\) to get \(5\).
• Finally, multiply these results: \(11 \cdot 5 = 55\).

Understanding this sequential process ensures the expression is evaluated correctly. Mastering arithmetic expressions boosts your confidence in tackling more complex equations.

Calculator Verification Techniques

Verifying your solutions with a calculator can be a valuable skill, especially when completing math problems that involve multiple steps. A calculator helps confirm your manual calculations and ensures accuracy.
To verify the solution of \((4+7) \cdot (8-3) = 55\) using a calculator:

• Enter the entire expression as given to ensure no step was overlooked.
• Double-check any intermediate steps by calculating individually, if needed.

Using a calculator not only helps in checking your work but can also serve as a learning tool. When you see the calculator providing the correct answer that matches your calculated result, it reinforces your understanding of the order of operations.
Keep in mind, while calculators are powerful, always attempt to solve problems manually first to enhance your mathematical skills. This will help build strong foundational knowledge, making you less reliant on electronic devices for basic computations.