Problem 31 - Elementary Algebra

Question

For the following problems, use the order of operations to find each value. $$(10+5)(10+5)-4(60-4)$$

Step-by-Step Solution

Answer

Answer: The value of the expression is 1.

Step 1: Calculate Inside Parentheses

First, we need to simplify the expressions inside the parentheses, which are \((10+5)\) and \((60-4)\). Let's do these calculations: \((10+5) = 15\) \((60-4) = 56\) Now we update the expression: $$15 \cdot 15 - 4 \cdot 56$$

Step 2: Perform Multiplication

Next, we will perform the multiplication operations: \(15 \cdot 15 = 225\) \(4 \cdot 56 = 224\) Now update the expression again: $$225 - 224$$

Step 3: Perform Subtraction

Finally, we will perform the subtraction operation: $$225 - 224 = 1$$ So, the value of the given expression is 1.

Key Concepts

Parentheses

In any mathematical expression, the order of operations is crucial to getting the correct result. The rule of parentheses takes priority, meaning that any calculations inside parentheses should be completed first. In the expression \((10+5)(10+5)-4(60-4)\), we focus on the two sets of parentheses:

• \((10+5)\) simplifies to 15
• \((60-4)\) simplifies to 56

By completing these operations first, we are setting a solid foundation for the remaining calculations. Once these parts of the expression are dealt with, we update the expression to: \[15 \cdot 15 - 4 \cdot 56\] This ensures clarity and accuracy as we proceed to the next steps.

Multiplication

The next step in solving the expression \(15 \cdot 15 - 4 \cdot 56\) is to handle the multiplication. Multiplication is executed before addition and subtraction in the order of operations. Here, we perform the following calculations:

• \(15 \cdot 15 = 225\)
• \(4 \cdot 56 = 224\)

Each multiplication operation transforms the expression into simpler components. It helps to write down each multiplication result before substituting back into the expression. Now, the expression simplifies to: \[225 - 224\] Proceeding in this manner ensures no steps are skipped and keeps the process straightforward.

Subtraction

Finally, we reach the subtraction stage in the expression \(225 - 224\). Subtraction is the last operation performed when evaluating expressions according to the order of operations. To complete this operation:

• Subtract 224 from 225, which results in: 1

This final step concludes the calculation, demonstrating how simply following the standard mathematical order of operations can solve even complex expressions. By understanding the sequence—parentheses first, followed by multiplication, and finishing with subtraction—we arrive at the correct solution: 1.