Problem 62 - Fundamentals of Mathematics

Question

Find each value. Check each result with a calculator. $$\frac{(2+1)^{3}+2^{3}+1^{10}}{6^{2}}-\frac{15^{2}-[2 \cdot 5]^{2}}{5 \cdot5^{2}}$$

Step-by-Step Solution

Answer

The result is 0.

Step 1: Simplify Numerators and Denominators

Start by simplifying each part of the expression. For the first fraction, calculate \((2+1)^{3} = 3^{3} = 27\), \(2^{3} = 8\), and \(1^{10} = 1\). Combine them to get \(27 + 8 + 1 = 36\). For the denominator, \(6^{2} = 36\). For the second fraction, the numerator is \(15^{2} - (2 \cdot 5)^{2} = 225 - 100 = 125\). The denominator is equal to \(5\cdot5^{2} = 5\cdot25 = 125\).

Step 2: Calculate Fractions

Divide the results from each numerator by its corresponding denominator: \(\frac{36}{36} = 1\) and \(\frac{125}{125} = 1\).

Step 3: Subtract the Fractions

Subtract the second term from the first: \(1 - 1 = 0\).

Step 4: Verify with a Calculator

Use a calculator to confirm the intermediate results: \(36 / 36 = 1\) and \(125 / 125 = 1\), and finally, \(1 - 1 = 0\). All calculations are correct, confirming the final result.

Key Concepts

Simplifying Expressions

Breaking down mathematical expressions into simpler forms is crucial for solving complex problems. Simplifying an expression involves removing parentheses, combining like terms, and reducing fractions where applicable.
This process not only makes calculations easier but also helps in the identification of patterns and further mathematical operations. Begin with looking at each term separately.

• Simplify terms inside brackets or parentheses first.
• Use operations such as addition, subtraction, multiplication, and division according to the order of operations, known as PEMDAS/BODMAS.

In our problem, each expression in both the numerators and denominators was simplified separately. This avoidance of overly complex expressions helped to achieve a clearer path to the solution.

Fractions

Understanding fractions is important when simplifying expressions involving division. Each fraction consists of a numerator, the number at the top, and a denominator, at the bottom.
Operations on fractions often require you to perform arithmetic on the numerator and denominator separately.

• To add or subtract fractions, ensure they have a common denominator.
• For multiplication, multiply the numerators together and the denominators together.
• Dividing fractions involves flipping the second fraction and multiplying.

In our exercise, fractions were formed by performing division of numerators by their corresponding denominators. Ensuring that each step maintains the balance between these two components is crucial to reaching the correct answer. This prevents errors and verifies solution accuracy.

Exponents

Exponents denote repeated multiplication of a base number. In expressions, they help represent large numbers in a compact form, simplifying calculations.
This is particularly helpful when dealing with powers such as squares and cubes.

• A base raised to an exponent like in \( 3^3 = 3 \times 3 \times 3 = 27 \) illustrates this concept.
• Common exponent rules include: \( a^m \cdot a^n = a^{m+n} \) and \( (a^m)^n = a^{m \cdot n} \).

In the given problem, exponents were used to calculate powers such as \( (2+1)^{3} \, 2^{3}, \, \text{and} \, 1^{10} \), which were crucial in simplifying the expressions step by step.

Arithmetic Operations

Arithmetic operations are basic computations used in math problems that include addition, subtraction, multiplication, and division. Each operation plays a role in transforming numbers following algebraic rules.
It's important to perform these operations in the correct order to ensure an accurate result. Known as the order of operations, it is represented by the acronym PEMDAS, which stands for Parentheses, Exponents, Multiplication, Division, Addition, Subtraction.

• Always solve operations inside parentheses first.
• Next, handle any exponents or powers.
• Then look to multiplication and division.
• Finally, perform addition and subtraction from left to right.

In the presented exercise, these basic operations guided the simplification and computation processes, leading to the final result. Following this order ensured that the operations were correctly executed, particularly when dealing with multiple steps in both numerators and denominators.