Solving (−1/8)³ × (-1)¹⁰⁵ × (8/5)² A Step-by-Step Guide

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In the realm of mathematics, problem-solving is an art form, a dance between logic and intuition. Today, we embark on a journey to unravel a seemingly complex equation: (−1/8)³ × (-1)¹⁰⁵ × (8/5)². This problem, a blend of exponents, fractions, and negative numbers, serves as an excellent exercise in applying fundamental mathematical principles. To conquer this challenge, we'll dissect each component, simplify it, and then piece the results together to arrive at the final answer. Let's embark on this mathematical adventure together, step by meticulous step.

Understanding the Building Blocks: A Detailed Breakdown

Before we dive into the solution, it's crucial to understand the individual components of the equation. This involves recognizing the order of operations, the properties of exponents, and how to handle negative numbers. Let's break down the equation piece by piece:

1. Exploring Exponents: The Power Within

Exponents represent repeated multiplication. For instance, x³ means x multiplied by itself three times (x × x × x). In our equation, we encounter two terms with exponents: (−1/8)³ and (8/5)². Let's delve deeper into each:

  • (−1/8)³: This term signifies that we need to multiply -1/8 by itself three times: (−1/8) × (−1/8) × (−1/8). When multiplying fractions, we multiply the numerators (the top numbers) and the denominators (the bottom numbers) separately. Also, remember the rule for multiplying negative numbers: a negative times a negative equals a positive, and a positive times a negative equals a negative. So, (-1/8)³ = (-1 × -1 × -1) / (8 × 8 × 8) = -1 / 512.
  • (8/5)²: This term means we multiply 8/5 by itself: (8/5) × (8/5). Again, we multiply the numerators and denominators: (8 × 8) / (5 × 5) = 64 / 25.

2. The Significance of Negative Numbers: Unveiling the Sign

Negative numbers add a layer of complexity, especially when combined with exponents. In our equation, we have (-1)¹⁰⁵. The exponent 105 tells us to multiply -1 by itself 105 times. A crucial rule to remember is that a negative number raised to an odd power results in a negative number, while a negative number raised to an even power yields a positive number. Since 105 is an odd number, (-1)¹⁰⁵ equals -1.

3. Fractions: The Parts of a Whole

Fractions represent parts of a whole and are written as a numerator over a denominator. In our equation, we have fractions like -1/8 and 8/5. Understanding how to multiply fractions is vital for solving the equation. As we saw earlier, when multiplying fractions, we multiply the numerators and the denominators separately.

4. Order of Operations: The Guiding Principle

The order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), dictates the sequence in which we perform mathematical operations. In our equation, we first need to address the exponents, then the multiplication.

Step-by-Step Solution: Putting the Pieces Together

Now that we've dissected the individual components, let's piece them together to solve the equation. We'll follow the order of operations, simplifying each term step-by-step:

  1. Simplify the exponents:

    • (−1/8)³ = -1/512
    • (-1)¹⁰⁵ = -1
    • (8/5)² = 64/25

  2. Substitute the simplified terms back into the equation:

    • The equation now becomes: (-1/512) × (-1) × (64/25)

  3. Perform the multiplication from left to right:

    • First, multiply (-1/512) by (-1): (-1/512) × (-1) = 1/512. Remember, a negative times a negative equals a positive.
    • Now, multiply the result (1/512) by (64/25): (1/512) × (64/25).

  4. Multiplying the fractions:

    • Multiply the numerators: 1 × 64 = 64
    • Multiply the denominators: 512 × 25 = 12800
    • The fraction becomes 64/12800

  5. Simplify the fraction:

    • Both the numerator and the denominator can be divided by 64.
    • 64 ÷ 64 = 1
    • 12800 ÷ 64 = 200
    • The simplified fraction is 1/200

Therefore, the solution to the equation (−1/8)³ × (-1)¹⁰⁵ × (8/5)² is 1/200.

Key Takeaways: Mastering the Fundamentals

This problem highlights the importance of understanding fundamental mathematical concepts: exponents, negative numbers, fractions, and the order of operations. By breaking down the problem into smaller, manageable parts, we can systematically arrive at the solution. Here are some key takeaways:

  • Exponents: Remember that exponents represent repeated multiplication. A negative number raised to an odd power is negative, and a negative number raised to an even power is positive.
  • Fractions: Multiplying fractions involves multiplying the numerators and the denominators separately. Always simplify the fraction to its lowest terms if possible.
  • Order of Operations: Adhering to the order of operations (PEMDAS) is crucial for accurate calculations.
  • Problem-Solving Strategy: Break down complex problems into smaller, easier-to-solve parts. This approach makes the problem less daunting and more manageable.

Practice Makes Perfect: Enhancing Your Mathematical Skills

Mathematical proficiency is built through practice. The more problems you solve, the more comfortable and confident you become. Try tackling similar problems with varying exponents, fractions, and negative numbers. Experiment with different problem-solving strategies and techniques. The journey of learning mathematics is a continuous process of exploration and discovery.

In Conclusion: The Elegance of Mathematical Solutions

The equation (−1/8)³ × (-1)¹⁰⁵ × (8/5)² initially appeared complex, but by carefully applying mathematical principles, we were able to unravel its intricacies and arrive at the solution. This problem exemplifies the beauty and elegance of mathematics – the power of logic, the precision of calculations, and the satisfaction of finding the right answer. Keep exploring, keep practicing, and keep unlocking the world of mathematics!