Problem 50 - Fundamentals of Mathematics

Question

Find each value. Check each result with a calculator. \((6+8) \cdot(5+2-4)\)

Step-by-Step Solution

Answer

The value is 42.

Step 1: Evaluate the First Parentheses

Identify and evaluate the expression within the first parentheses: \[(6 + 8)\]Calculate: \[6 + 8 = 14\]

Step 2: Evaluate the Second Parentheses

Identify and evaluate the expression within the second parentheses: \[(5 + 2 - 4)\]Calculate:\[5 + 2 - 4 = 3\]

Step 3: Multiply the Results from the Parentheses

Use the results from Step 1 and Step 2 to perform the multiplication: \[14 \cdot 3\]Calculate:\[14 \times 3 = 42\]

Step 4: Verify the Result with a Calculator

Input the initial expression into a calculator to verify the result:\[(6 + 8) \times (5 + 2 - 4)\]The calculator should also display the result as 42, confirming the calculation.

Key Concepts

Parentheses

In mathematical expressions, parentheses are used to explicitly indicate which operations should be performed first. They help in organizing calculations and ensuring clarity of the intended order. In our problem, we see parentheses surrounding two separate expressions:

• \((6 + 8)\)
• \((5 + 2 - 4)\)

The purpose of these parentheses is to ensure that the operations inside them—addition and subtraction—are completed before moving on to other operations outside the parentheses.
Start by tackling each expression individually. For the first group, \(6 + 8\), add the two numbers to get 14.
For the second group, \(5 + 2 - 4\), first add 5 and 2 to get 7, then subtract 4 to reach 3. These parentheses guided us to prioritize their enclosed calculations before applying any other operations.

Multiplication

After solving the expressions within the parentheses, the next step is to perform multiplication. The multiplication in our expression comes after evaluating both sets of parentheses:

• First group's result: 14
• Second group's result: 3

Now, these results become the factors in the multiplication step. Multiply the number 14 by 3.
Simply put, \(14 \times 3\) equals 42.
This multiplication follows the basic property of multiplication where two numbers (factors) are combined to give a single number (product). Understanding this operation is crucial since multiplication often follows after calculating with parentheses.

Calculator Verification

After completing manual calculations, it is important to verify your results using a calculator. This serves as a double-check to confirm your arithmetic is correct, ensuring that no mistakes were made during the step-by-step process.
For our expression \((6 + 8) \times (5 + 2 - 4)\), input each part into the calculator exactly as shown, ensuring parentheses are included.
Calculators follow the order of operations themselves, so when you input the expression, it should automatically evaluate inside the parentheses first. It will then proceed to the multiplication of the resulting numbers: \(14 \times 3 = 42\).
If the displayed result is the same as your manual calculation, you can be confident in the accuracy of your work.