Equivalent Expression For (5/8)^(-1)(0.42)^2 A Step-by-Step Solution

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Hey there, math enthusiasts! Today, we're diving deep into the fascinating world of equivalent expressions. Our mission? To unravel the expression (5/8)(-1)(0.42)2 and discover its many faces. Think of it as a mathematical makeover โ€“ same value, different looks! Whether you're a student tackling algebra, a curious mind exploring mathematical concepts, or just someone who loves a good equation, this guide is for you. We'll break down each step, explain the underlying principles, and ensure you not only understand the solution but also the 'why' behind it. So, buckle up, and let's embark on this mathematical adventure together!

Deconstructing the Expression: A Step-by-Step Analysis

Our journey begins with the expression (5/8)(-1)(0.42)2. At first glance, it might seem like a jumble of fractions, exponents, and decimals. But don't worry, guys! We're going to dissect it piece by piece, transforming it into simpler, more manageable forms. The beauty of mathematics lies in its precision and the ability to manipulate expressions while preserving their inherent value. This process is not just about finding an answer; it's about understanding the many ways to represent the same mathematical truth.

Tackling the Negative Exponent: Flipping the Fraction

The first part of our expression, (5/8)^(-1), features a negative exponent. Remember the rule: a negative exponent means we need to take the reciprocal of the base. In simpler terms, we flip the fraction! So, (5/8)^(-1) becomes (8/5). This transformation is crucial because it gets rid of the negative exponent, making the expression easier to work with. This step highlights a fundamental property of exponents, allowing us to rewrite expressions in more convenient forms. Now, instead of dealing with a potentially cumbersome fraction raised to a negative power, we have a straightforward fraction, 8/5. This is a significant simplification, setting the stage for the next steps in our journey.

Decoding the Decimal: Converting to a Fraction

Next up, we have (0.42)^2. Dealing with decimals can sometimes be tricky, so let's convert 0.42 into a fraction. 0.42 is equivalent to 42/100. Now, we can simplify this fraction by finding the greatest common divisor (GCD) of 42 and 100, which is 2. Dividing both the numerator and the denominator by 2, we get 21/50. This conversion is a game-changer, transforming a decimal into a fraction, which aligns perfectly with the other part of our expression. Working with fractions often allows for easier manipulation and simplification, especially when dealing with exponents and multiplication.

Squaring the Fraction: Multiplying by Itself

Now that we've converted 0.42 to 21/50, we need to square it. Squaring a fraction means multiplying it by itself: (21/50)^2 = (21/50) * (21/50). Multiplying the numerators, 21 * 21 = 441. Multiplying the denominators, 50 * 50 = 2500. So, (21/50)^2 = 441/2500. This step showcases the mechanics of squaring fractions, a fundamental operation in algebra and beyond. By squaring the fraction, we're essentially finding the area of a square with sides of length 21/50, a visual representation that can aid in understanding.

Bringing It All Together: The Grand Finale of Simplification

We've transformed (5/8)^(-1) into 8/5 and (0.42)^2 into 441/2500. Now, it's time for the grand finale โ€“ multiplying these two simplified expressions together: (8/5) * (441/2500). This multiplication will give us a single fraction that represents the value of our original expression. Multiplying fractions is straightforward: we multiply the numerators and the denominators separately. So, 8 * 441 = 3528, and 5 * 2500 = 12500. Our expression now looks like this: 3528/12500.

Reducing to Lowest Terms: Finding the Simplest Form

Our final fraction, 3528/12500, is a valid representation of the original expression, but it's not in its simplest form. To simplify it, we need to find the greatest common divisor (GCD) of 3528 and 12500 and divide both the numerator and the denominator by it. The GCD of 3528 and 12500 is 4. Dividing both by 4, we get 3528 / 4 = 882 and 12500 / 4 = 3125. So, our simplified fraction is 882/3125. This reduction is crucial for presenting the answer in its most elegant and concise form. Simplified fractions are easier to compare, understand, and work with in further calculations.

Exploring Decimal Equivalents: Bridging the Gap

While the fraction 882/3125 is a perfectly valid answer, sometimes it's helpful to express it as a decimal. To convert a fraction to a decimal, we simply divide the numerator by the denominator. So, 882 รท 3125 = 0.28224. This decimal representation gives us another perspective on the value of the expression, allowing us to see it in a different light. Converting between fractions and decimals is a fundamental skill in mathematics, enabling us to choose the representation that best suits the context of the problem.

Equivalent Expressions: A World of Possibilities

So, what have we achieved? We started with (5/8)(-1)(0.42)2 and transformed it into several equivalent expressions: 8/5 * 441/2500, 3528/12500, 882/3125, and 0.28224. Each of these expressions represents the same value, but they look different and might be more useful in different situations. This is the essence of equivalent expressions โ€“ different forms, same value. Understanding equivalent expressions is a cornerstone of mathematical fluency, allowing us to manipulate equations and solve problems with greater ease and flexibility.

Why This Matters: The Power of Equivalent Expressions

Why go through all this trouble to find equivalent expressions? Because it unlocks a world of possibilities in mathematics! Equivalent expressions allow us to: Simplify complex equations, making them easier to solve. Compare different mathematical quantities, even when they're presented in different forms. Choose the most convenient form for a particular calculation. Gain a deeper understanding of the relationships between different mathematical concepts. The ability to manipulate expressions and find equivalents is not just a skill; it's a superpower in the world of math!

Conclusion: Mastering the Art of Transformation

We've journeyed through the intricacies of the expression (5/8)(-1)(0.42)2, transforming it into a variety of equivalent forms. We tackled negative exponents, converted decimals to fractions, squared fractions, multiplied, simplified, and even explored decimal equivalents. Along the way, we've reinforced fundamental mathematical principles and gained a deeper appreciation for the power of equivalent expressions. So, next time you encounter a complex expression, remember the tools and techniques we've discussed. Embrace the challenge, break it down, and transform it! You've got this, guys!

What is an equivalent expression for (5/8)(-1)(0.42)2? Let's break down this mathematical problem step-by-step to find the solution. This article will guide you through the process of simplifying the expression, converting decimals to fractions, and understanding the properties of exponents. Whether you're a student, a math enthusiast, or just someone looking to brush up on your algebra skills, this comprehensive guide will provide you with a clear and concise explanation. We'll cover each step in detail, ensuring you grasp the fundamental concepts and can apply them to similar problems. So, grab your calculator (or your brain!), and let's dive into the world of equivalent expressions!

Can you find an equivalent expression for (5/8)(-1)(0.42)2? This mathematical puzzle combines fractions, exponents, and decimals, offering a fantastic opportunity to flex your math muscles. In this article, we'll meticulously dissect the expression, applying the rules of exponents and fraction manipulation to arrive at a simplified solution. We'll start by addressing the negative exponent, then convert the decimal to a fraction, and finally, perform the necessary calculations. Our goal is not just to provide the answer, but to empower you with the knowledge and skills to tackle similar challenges. Get ready for a journey into the heart of algebraic simplification!