Problem 33 - Fundamentals of Mathematics

Question

Use the order of operations to determine each value. \(\frac{6^{2}-1}{5 \cdot 7}-\frac{49+7}{2 \cdot 7}\)

Step-by-Step Solution

Answer

The value of the expression is -3.

Step 1: Evaluate Exponents

First, calculate the exponent in the expression. Here, evaluate the square in the numerator of the first fraction: \(6^2 = 36\). So the expression becomes \(\frac{36-1}{5 \cdot 7} - \frac{49+7}{2 \cdot 7}\).

Step 2: Perform Addition and Subtraction inside Numerators

Next, perform the subtraction and addition operations inside the numerators of the fractions:\(36 - 1 = 35\) and \(49 + 7 = 56\). The expression now simplifies to: \(\frac{35}{5 \cdot 7} - \frac{56}{2 \cdot 7}\).

Step 3: Multiply Denominators

Now, calculate the product in the denominators of each fraction:\(5 \cdot 7 = 35\) and \(2 \cdot 7 = 14\). The expression is updated to: \(\frac{35}{35} - \frac{56}{14}\).

Step 4: Simplify Fractions

Simplify each fraction:\(\frac{35}{35} = 1\) and \(\frac{56}{14} = 4\). The expression simplifies to:\(1 - 4\).

Step 5: Final Calculation

Perform the final subtraction:\(1 - 4 = -3\).

Key Concepts

Understanding Evaluating Expressions

When evaluating mathematical expressions, it is crucial to follow a systematic order to ensure you arrive at the correct result. This process involves looking at different parts of the expression and calculating them step by step, often beginning with solving any operations involving exponents, then moving onto other arithmetic operations like multiplication, division, addition, and subtraction.

Evaluating expressions involves:

• Identifying and solving any exponents first, as they have higher precedence compared to other arithmetic operations.
• Performing any operations within parentheses, although there are none in this exercise.
• Addressing multiplication and division from left to right.
• Finalizing with addition and subtraction from left to right.

Understanding such rules is essential in evaluating expressions correctly and efficiently. This meticulous approach ensures that every part of the expression is calculated in the correct sequence, leading to accurate results.

The Art of Simplifying Fractions

Simplifying fractions is an important skill in math that helps to make complex expressions easier to handle. When you simplify a fraction, you reduce it to its simplest form by dividing both the numerator and the denominator by their greatest common divisor (GCD).

To simplify fractions:

• Calculate the GCD of the numerator and denominator.
• Divide both the numerator and the denominator by the GCD.
• Ensure that the result is a fraction where the numerator and denominator can't be divided by the same number anymore (except for 1).

In the provided exercise, after calculating the products in the denominators, the fractions are simplified by recognizing simple integer divisions, making these fractions easier to handle for further arithmetic operations.

Mastering Arithmetic Operations

Arithmetic operations are the foundation of many mathematical problem-solving strategies. These operations include addition, subtraction, multiplication, and division. When combined carefully and in the correct order, they help solve expressions and equations effectively.

Key aspects to remember:

• Exponents are evaluated before any other operation to ensure the integrity of the expression's value.
• Follow the left-to-right rule for multiplication and division, giving them equal priority.
• Finish with addition and subtraction, also processed from left to right.

By mastering these operations, students can build confidence and competence in solving both basic and complex mathematical problems.

Effective Mathematical Problem-Solving

Mathematical problem-solving requires a blend of understanding concepts and applying techniques in a structured manner. The order of operations is a crucial strategy that helps students simplify and solve mathematical expressions accurately.

Problem-solving strategies often involve:

• Breaking down the problem into smaller, more manageable steps.
• Identifying the relevant operations and concepts that need to be applied.
• Sequentially processing the operations according to mathematical rules such as the order of operations.
• Checking the results obtained at each stage to avoid errors.

These strategies not only simplify the problem-solving process but also enhance analytical thinking and ensure that students arrive at the right solutions consistently. With regular practice, these techniques become intuitive, leading to greater mathematical proficiency.