Problem 6 - Prealgebra

Question

Find the value of each expression. $$2[3+7(4)]$$

Step-by-Step Solution

Answer

The value of the expression is 62.

Step 1: Simplify Inside the Parentheses

First, we perform the operations inside the parentheses. Here, we have \(3 + 7(4)\). According to the order of operations, we first calculate the multiplication within the brackets. So, compute \(7 \times 4\), which equals 28.

Step 2: Add the Result

After finding the product, we replace the multiplication with the result in the expression: \(3 + 28\). Now, add these two numbers together: \(3 + 28 = 31\).

Step 3: Multiply with the Outside Number

Next, we return to the original expression \(2[3 + 7(4)]\). After simplifying the inside, we found it to be 31. Now, multiply this result by the 2 outside the bracket: \(2 \times 31\).

Step 4: Final Calculation

Now compute the final multiplication: \(2 \times 31 = 62\). This gives us the value of the entire expression.

Key Concepts

Arithmetic

Arithmetic involves the basic operations of math, including addition, subtraction, multiplication, and division. Each operation is a fundamental building block for more complex math concepts. Understanding arithmetic is crucial as it forms the foundation for solving equations and expressions.

• Addition: Combines two or more numbers to get a total or sum. Example: 3 + 4 = 7.
• Multiplication: Essentially repeated addition. Instead of adding a number several times, multiplication condenses the process. For example, 4 multiplied by 3 (4 x 3) means adding 4 three times (4 + 4 + 4 = 12).

Mastering these operations enables you to follow the correct sequence of solving any mathematical problem. Be it simple arithmetic or complex algebra, these operations guide you through to the solution.

Parentheses in Math

Parentheses, often seen as brackets or grouping symbols, are vital in mathematics. They dictate the order in which calculations should be performed. Whenever you come across parentheses in an expression, the operations inside them need to be completed first.For example, in the expression \(2[3 + 7(4)]\), you must first address the calculations inside the parentheses. This is a part of the order of operations rule, often remembered by the acronym PEMDAS:

• P: Parentheses
• E: Exponents (i.e. powers and roots, etc.)
• M/D: Multiplication and Division (from left to right)
• A/S: Addition and Subtraction (from left to right)

By performing the steps in the correct sequence, parentheses ensure that the intended operations are completed accurately, leading to the correct result.

Multiplication and Addition

Multiplication and addition are operations frequently used together in expressions and equations. The sequence in which they are performed is crucial and is guided by the order of operations.Multiplication comes before addition in terms of priority. For instance, when looking at the expression \(3 + 7(4)\), you multiply first: calculate \(7 \times 4 = 28\), and then add the result to 3, making it \(3 + 28 = 31\).This hierarchy of operations demonstrates the integral role of multiplication in simplifying expressions before moving on to addition:

• Multiplication takes precedence over addition, allowing for more efficient simplification of expressions.
• Addition is then performed with the multiplication result, yielding a complete solution when combined with other operations.

Consistency in following this order helps in producing accurate results in calculations, ensuring the reliability of mathematical solutions.