Problem 59 - Prealgebra

Question

Find the value of the expression \(4 \cdot(8+9)+6\).

Step-by-Step Solution

Answer

The value of the expression is 74.

Step 1: Simplify the Expression Inside the Parentheses

Start by simplifying the expression inside the parentheses. In the expression \(8+9\), add the two numbers together: \(8+9=17\).

Step 2: Multiply by the Value Outside the Parentheses

Next, take the result from Step 1 and multiply it by the number outside the parentheses. This gives us \(4 \cdot 17\). Multiply these numbers: \(4 \times 17 = 68\).

Step 3: Add the Remaining Number

Finally, take the result from Step 2 and add the remaining number, which is \(6\). Add \(68 + 6 = 74\).

Key Concepts

Understanding Parentheses in Mathematics

When dealing with mathematical expressions, parentheses are used to prioritize the operations within them. To simplify expressions correctly, always start with calculations inside parentheses. In our original exercise, for instance, we have the expression \((8+9)\). Before carrying out any other operations, you should solve what's inside the parentheses first. This is an essential rule of the order of operations. It helps avoid any confusion in multi-step calculations.

When you see parentheses:

• Focus on them first before dealing with other operations outside.
• Ensure you perform all calculations inside before moving on.
• Remember, parentheses help clarify which operations should be prioritized.

Keeping these points in mind will enhance your ability to parse and solve arithmetic problems accurately.

The Role of Addition in Expressions

Addition is one of the fundamental operations in arithmetic. It is used to combine numbers. When you add two numbers, like in our example with \(8+9\), you're finding their total or sum. Addition is typically one of the last operations you perform in a simplified expression, unless it’s within parentheses.

Here is how to handle addition in expressions:

• If addition is within parentheses, perform it immediately, following the order of operations.
• If the addition is outside other operations like multiplication, carry it out afterward.
• Addition is commutative meaning \(a + b = b + a\), which helps in reordering sums when needed.

Understanding how to prioritize addition helps accurately compute totals and makes tackling complex expressions much simpler.

Mastering Multiplication in Order of Operations

Multiplication is another cornerstone operation in math, which is performed after dealing with parentheses, depending on its placement relative to other operations. In the expression from our exercise, \(4 \cdot (8+9)+6\), we handled the addition within the parentheses first.

After resolving \(8+9\) to 17, multiplication followed with \(4 \times 17\) before addition.

Consider these guidelines for multiplication:

• After parentheses, address multiplication before addition or subtraction outside.
• Ensure you apply the multiplication across all terms as expressed.
• Multiplication distributes over addition, ensuring terms within parentheses are accurately expanded if needed.

Adhering to these principles assists in properly simplifying mathematical expressions, ensuring your calculations are both accurate and efficient.