Solving (0.4)^-2 + (0.6)^-2 A Step-by-Step Mathematical Guide
Introduction
In the realm of mathematics, seemingly complex equations often yield to the application of fundamental principles. This article delves into the resolution of the expression (0.4)^-2 + (0.6)^-2, providing a step-by-step breakdown to illuminate the underlying concepts and arrive at the correct answer. We will explore the properties of exponents, fractions, and basic arithmetic operations to demystify the problem and make it accessible to learners of all levels. Whether you're a student grappling with exponents or simply seeking to enhance your mathematical prowess, this guide will equip you with the knowledge and skills to tackle similar challenges with confidence.
Understanding Negative Exponents
The cornerstone of solving this equation lies in comprehending the nature of negative exponents. A negative exponent signifies the reciprocal of the base raised to the positive value of the exponent. In simpler terms, x^-n is equivalent to 1 / x^n. This principle is crucial for transforming the given expression into a more manageable form. Applying this rule to our problem, (0.4)^-2 becomes 1 / (0.4)^2 and (0.6)^-2 transforms into 1 / (0.6)^2. By rewriting the terms with positive exponents, we pave the way for further simplification.
Converting Decimals to Fractions
To facilitate calculations, it's often beneficial to convert decimals into fractions. The decimal 0.4 can be expressed as the fraction 4/10, which simplifies to 2/5. Similarly, 0.6 translates to 6/10, simplifying to 3/5. By representing the decimals as fractions, we can leverage the rules of fraction arithmetic to our advantage. This conversion not only simplifies the calculations but also provides a clearer understanding of the numerical relationships involved. The fractions 2/5 and 3/5 will now be the basis for our exponentiation and subsequent addition.
Step-by-Step Solution
Step 1: Applying the Negative Exponent Rule
As established earlier, we begin by applying the negative exponent rule to rewrite the expression. This transforms (0.4)^-2 + (0.6)^-2 into 1 / (0.4)^2 + 1 / (0.6)^2. This step is pivotal as it eliminates the negative exponents, making the expression easier to work with. By understanding this fundamental rule, we can confidently handle any term raised to a negative power.
Step 2: Converting Decimals to Fractions (Revisited)
We reiterate the conversion of decimals to fractions, expressing 0.4 as 2/5 and 0.6 as 3/5. This gives us 1 / (2/5)^2 + 1 / (3/5)^2. The use of fractions allows for a more precise representation of the numbers and simplifies the process of squaring and finding reciprocals.
Step 3: Squaring the Fractions
Next, we square the fractions within the parentheses. (2/5)^2 equals (2/5) * (2/5) = 4/25, and (3/5)^2 equals (3/5) * (3/5) = 9/25. Our expression now becomes 1 / (4/25) + 1 / (9/25). Squaring the fractions involves multiplying the numerators and denominators separately, a straightforward process that yields the results needed for the next step.
Step 4: Finding the Reciprocals
Dividing by a fraction is the same as multiplying by its reciprocal. Therefore, 1 / (4/25) is equivalent to 1 * (25/4) = 25/4, and 1 / (9/25) is equivalent to 1 * (25/9) = 25/9. This step utilizes the principle of reciprocals, a fundamental concept in fraction division. By inverting the fractions, we transform the division problem into a multiplication problem, making it easier to solve.
Step 5: Adding the Fractions
To add the fractions 25/4 and 25/9, we need to find a common denominator. The least common multiple of 4 and 9 is 36. We convert 25/4 to (25 * 9) / (4 * 9) = 225/36 and 25/9 to (25 * 4) / (9 * 4) = 100/36. Now we can add the fractions: 225/36 + 100/36 = 325/36. Finding a common denominator is a crucial step in adding or subtracting fractions, ensuring that we are adding like terms.
Final Answer and Conclusion
Therefore, (0.4)^-2 + (0.6)^-2 = 325/36. This result corresponds to option D in the given choices. By systematically applying the rules of exponents, fractions, and arithmetic operations, we have successfully solved the equation. This exercise highlights the importance of understanding fundamental mathematical principles and their application in problem-solving. The ability to manipulate exponents and fractions is a valuable skill in various mathematical contexts.
Key Takeaways
- Negative exponents indicate reciprocals.
- Converting decimals to fractions can simplify calculations.
- Dividing by a fraction is the same as multiplying by its reciprocal.
- Adding fractions requires a common denominator.
By mastering these key concepts, you can confidently tackle a wide range of mathematical problems. Practice and consistent application of these principles will further solidify your understanding and enhance your problem-solving abilities.
Additional Practice Problems
To reinforce your understanding, consider attempting these additional practice problems:
- (0.2)^-2 + (0.8)^-2 =
- (1/3)^-2 + (1/4)^-2 =
- (0.5)^-2 - (0.25)^-2 =
Working through these problems will provide further practice in applying the concepts discussed in this article. Remember to break down each problem into manageable steps and utilize the principles of exponents, fractions, and arithmetic operations.
By diligently practicing and applying these concepts, you can build a strong foundation in mathematics and confidently approach complex problems. The world of mathematics is filled with fascinating challenges, and with the right tools and understanding, you can unlock its many secrets.
