Problem 13 - Fundamentals of Mathematics

Question

Determine the value of each expression. \(3 \cdot \frac{8^{2}-2 \cdot 3^{2}}{5^{2}-2} \cdot \frac{6^{3}-4 \cdot 5^{2}}{29}\)

Step-by-Step Solution

Answer

The value of the expression is 24.

Step 1: Simplify the Numerator of the First Fraction

First, evaluate the expression in the numerator of the first fraction: \[8^{2} - 2 \cdot 3^{2}\].Calculate each term: \[8^{2} = 64\] and \[2 \cdot 3^{2} = 2 \cdot 9 = 18\].Subtract the results: \[64 - 18 = 46\].Thus, the numerator of the first fraction simplifies to 46.

Step 2: Simplify the Denominator of the First Fraction

Now calculate the denominator of the first fraction: \[5^{2} - 2\].Calculate \[5^{2} = 25\].Subtract: \[25 - 2 = 23\].Hence, the denominator simplifies to 23.

Step 3: Evaluate the First Fraction

Divide the simplified numerator by the denominator: \[\frac{46}{23} = 2\].So, the first fraction simplifies to 2.

Step 4: Simplify the Numerator of the Second Fraction

Next, evaluate the expression in the numerator of the second fraction:\[6^{3} - 4 \cdot 5^{2}\].Calculate each term: \[6^{3} = 216\] and \[4 \cdot 5^{2} = 4 \cdot 25 = 100\].Subtract the results: \[216 - 100 = 116\].Thus, the numerator of the second fraction simplifies to 116.

Step 5: Evaluate the Second Fraction

Now divide the simplified numerator by the given denominator, 29:\[\frac{116}{29} = 4\].So, the second fraction simplifies to 4.

Step 6: Calculate the Product

Now multiply the results from the previous steps and the initial coefficient:Calculate \[3 \cdot 2 \cdot 4\].First multiply \[3 \cdot 2 = 6\], then multiply \[6 \cdot 4 = 24\].Thus, the entire expression simplifies to 24.

Key Concepts

Arithmetic Expressions

Arithmetic expressions are mathematical phrases that use numbers, operators, and sometimes parentheses to represent a calculation. In the given exercise, the arithmetic expression is \[3 \cdot \frac{8^{2}-2 \cdot 3^{2}}{5^{2}-2} \cdot \frac{6^{3}-4 \cdot 5^{2}}{29}\].Here's a breakdown of components in arithmetic expressions:

• Numbers: Basic building blocks of the expression, like 3, 8, 5, etc.
• Operators: Symbols that represent actions, such as addition, subtraction, multiplication (\(\cdot\)), and division \((\div)\).
• Parentheses: Used to group parts of the expression and show which parts should be evaluated first.

Arithmetic expressions follow the rules of the order of operations, typically remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division (left-to-right), Addition and Subtraction (left-to-right)). This ensures the expression is evaluated in the correct sequence to yield the right result.Understanding how to manipulate arithmetic expressions is critical in solving complex calculations and equations. They form the basis for most mathematical problem-solving techniques.

Fraction Simplification

Fraction simplification is the process of reducing fractions to their simplest form, which means having the smallest possible numerator and denominator that maintain the same value.In our original exercise, we encountered fractions such as:1. \(\frac{8^{2}-2 \cdot 3^{2}}{5^{2}-2}\)2. \(\frac{6^{3}-4 \cdot 5^{2}}{29}\)To simplify a fraction, you need to evaluate the numerator and the denominator separately. Then, divide them to find the simplest form:

• First, calculate the expressions in the numerator and denominator.
• Next, simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor when possible.

For our problem:
- The simplified first fraction \(\frac{46}{23}\) becomes 2 because 23 goes into 46 twice.
- The second fraction \(\frac{116}{29}\) simplifies to 4 since 29 goes into 116 four times.Fraction simplification plays a vital role in making problems easier to solve, particularly when multiple fractions are involved. Recognizing patterns and breaking down numbers help to simplify seemingly complicated expressions, making them more manageable.

Mathematical Evaluation

Mathematical evaluation involves using rules and steps to find the value of an arithmetic expression. It requires careful execution of mathematical operations, respecting the order of operations.In the exercise, we evaluated the expression\[3 \cdot 2 \cdot 4\].The steps to evaluate this included:

• Calculation: Each component expression was simplified, such as evaluating powers and performing arithmetic operations within fractions.
• Combination: With simple numbers, the final multiplication was performed sequentially.
• Result Verification: This involves confirming every calculation to ensure it's correctly executed, resulting in the final value.

For instance, combining our simplified fraction values (2 and 4) with an initial multiplier (3) involved careful steps:- Multiply 3 by 2 to get 6.
- Then, multiply 6 by 4 to reach 24, the final result.Mathematical evaluation is essential for ensuring that expressions are interpreted and solved accurately. It helps students follow a process that systematically breaks down a problem using both straightforward and complex mathematical operations.