Problem 100 - Prealgebra

Question

Simplify. $$6 \cdot 8+3(4-1)$$

Step-by-Step Solution

Answer

The expression simplifies to 57.

Step 1: Solve Inside the Parentheses

The expression inside the parentheses is \(4 - 1\). Simplify this to find that \(4 - 1 = 3\). Therefore, the expression becomes \(6 \cdot 8 + 3 \cdot 3\).

Step 2: Perform the Multiplication

Now calculate each multiplication separately. First, compute \(6 \cdot 8\) which equals 48. Then, compute \(3 \cdot 3\) which equals 9. This results in the expression \(48 + 9\).

Step 3: Add the Products

Finally, add the products obtained from the previous multiplications: \(48 + 9 = 57\). Therefore, the simplified form of the expression is 57.

Key Concepts

Simplifying Expressions

Simplifying expressions in mathematics involves reducing complexity while maintaining the original value of the expression. It is often about performing mathematical operations in the correct order to make an expression easier to understand or solve. Understanding how to simplify expressions requires knowing the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).

• Start with Parentheses: Simplifying usually begins with solving expressions inside parentheses. Doing this first ensures we respect the implicit priority given to what's inside the parentheses.
• Move Next to Multiplication and Division: Once the operations inside parentheses are complete, we move on to any multiplication and division. It's important to tackle these from left to right as they appear in the expression.
• Finish with Addition and Subtraction: The final step in simplification is performing any addition or subtraction, also from left to right.

By following this structured approach, we can simplify even complex expressions effectively.
Reducing expressions in this manner results in a more straightforward numerical answer, often revealing the inherent value in the expression.

Parentheses in Math

Parentheses in math are symbols used to group parts of expressions to indicate that the operations within them should be performed before the rest of the expression. They play a crucial role in ensuring calculations are done in the correct order.When simplifying expressions, parentheses guide us in calculating a portion of the expression separately.
Consider this: Whenever you see a problem like \[6 \cdot 8 + 3(4-1)\], the part inside the parentheses, \[4 - 1\], must be simplified first, resulting in \[3\].This simplification changes the expression to \[6 \cdot 8 + 3 \cdot 3\].

• Parentheses are a priority: This means any calculation within them should come first, even before multiplication or division.
• Establish clear calculation steps: Solving inside the parentheses first helps create a clear order in which other operations take place.

Using parentheses is a powerful technique that helps manage complex expressions by controlling the order of operations and minimizing mistakes.

Multiplication and Addition

Multiplication and addition are two of the basic arithmetic operations but are calculated in a specific order when simplifying expressions.In the math expression \[6 \cdot 8 + 3 \cdot 3\],once any parentheses are resolved, we handle multiplication before addition. Each product is calculated separately:

• Multiplication: First, determine the product of each multiplication, such as \[6 \cdot 8 = 48\]and \[3 \cdot 3 = 9\].
• Addition: Once the multiplication is complete, we sum the products. For instance, \[48 + 9 = 57\].

Handling these operations in their proper order is key.
Always multiply before adding when no other operations (such as division or subtraction) interfere.
This approach ensures accuracy in arriving at the final, simplified value of the expression.