Problem 71 - Intermediate Algebra

Question

Simplify each of the numerical expressions. $$ [7+2 \cdot 3 \cdot 5-5] \div 8 $$

Step-by-Step Solution

Answer

The simplified expression is 4.

Step 1: Address Multiplication Inside Brackets

First, solve the multiplication inside the brackets: \(2 \cdot 3 \cdot 5\). Compute the product: \(2 \cdot 3 = 6\) and then \(6 \cdot 5 = 30\).

Step 2: Simplify the Expression Inside Brackets

Insert the product from Step 1 back into the expression: \([7 + 30 - 5]\).

Step 3: Perform Additions and Subtractions Inside Brackets

Calculate the expressions inside the brackets: \(7 + 30 = 37\), then \(37 - 5 = 32\). The expression now is \([32]\).

Step 4: Divide by 8

Divide the simplified number by 8. So, \(32 \div 8 = 4\).

Key Concepts

Numerical Expressions

Numerical expressions are a fundamental part of mathematics as they involve combinations of numbers and operations that represent a particular value. These expressions can include addition, subtraction, multiplication, division, and even exponents or roots. A crucial aspect of working with numerical expressions is understanding how to interpret and simplify them correctly.

When dealing with numerical expressions, it is important to follow the order of operations—often remembered through the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). This standard practice ensures that calculations are completed in a consistent and correct manner.

In evaluating our given numerical expression \[ [7+2 \cdot 3 \cdot 5-5] \div 8 \], the operations inside the brackets must be determined first, as per PEMDAS. This step-by-step approach helps to avoid errors and ensures the correct result is obtained for any given numerical problem.

Arithmetic Simplification

Arithmetic simplification involves breaking down a complex expression into simpler parts to make calculations easier. This process includes reducing expressions step by step until they are in the simplest form, often by removing parentheses and combining like terms.

In our example, we start by simplifying the operations contained within the brackets of the expression \([7+2 \cdot 3 \cdot 5-5]\). This involves first handling the multiplication:

• Calculate \(2 \cdot 3\) to get 6
• Then, multiply 6 by 5 to get 30

This result replaces the original multiplication, allowing us to proceed with addition and subtraction.

Once calculated, we simplify further to reach a single numerical value. Here, \([7+30-5]\) becomes \([32]\) after performing the arithmetic operations. Ensuring each step is carefully simplified leads to accurate final results.

Mathematical Problem Solving

Mathematical problem solving involves applying mathematical techniques and knowledge to solve real-world or theoretical problems. This skill is crucial not only in academics but also in everyday situations. To solve problems effectively, one must be familiar with various mathematical concepts and operations.

In solving \([7+2 \cdot 3 \cdot 5-5] \div 8 \), we apply multiple concepts:

• Ordering the operations appropriately, using the order of operations (PEMDAS)
• Breaking down complex expressions through simplification

Finally, the division step must be performed carefully. Once the expression within the brackets is simplified to 32, dividing by 8 gives us the result, 4.

This structured approach to solving such mathematical expressions can be applied across various problems, providing clear pathways to solutions. By practicing these methods regularly, students will improve their problem-solving skills, fostering both confidence and competence in mathematics.