Evaluate (-1 2/3)^2: Step-by-Step Solution

Guys, let's dive into this mathematical problem together! We're going to evaluate the expression $\left(-1 \frac{2}{3}\right)^2$. It might seem a bit daunting at first, but don't worry, we'll break it down step by step so you can easily understand it. This isn't just about getting the right answer; it's about understanding the process, which is super important for tackling more complex math problems in the future.

Understanding the Expression

First, let's understand the expression $\left(-1 \frac{2}{3}\right)^2$. What does it actually mean? Well, it's essentially saying, “Take the number negative one and two-thirds, and then square it.” Squaring a number just means multiplying it by itself. So, we're looking at: $\left(-1 \frac{2}{3}\right) \times \left(-1 \frac{2}{3}\right)$. The key here is that we're dealing with a mixed number and we need to handle that negative sign carefully. Before we can square it, we need to convert the mixed number into an improper fraction. This will make the multiplication much easier and less prone to errors. Trust me, this is a crucial step that will save you a lot of headaches! Improper fractions give us a clear numerator and denominator to work with, which is exactly what we need when we're squaring something. Ignoring this step can lead to confusion and incorrect calculations, especially when negative signs are involved. This foundational understanding is what helps us tackle more complex problems later on. So, let's solidify this: squaring means multiplying by itself, and for mixed numbers, converting to improper fractions is our best friend.

Converting Mixed Numbers to Improper Fractions

Okay, so how do we convert this mixed number, $-1 \frac{2}{3}$, into an improper fraction? Remember the rule? We multiply the whole number part (1) by the denominator (3) and then add the numerator (2). This gives us the new numerator, and we keep the same denominator. So, (1 * 3) + 2 = 5. This means that $1 \frac{2}{3}$ as an improper fraction is $\frac{5}{3}$. But, don't forget about that negative sign! Our original number was negative, so the improper fraction is $-\frac{5}{3}$. This is a super common step in algebra and arithmetic, and it's one of those things that once you get the hang of, becomes second nature. Why do we do this? Because multiplying fractions is way easier than multiplying mixed numbers! Think about it: you just multiply the numerators and then multiply the denominators. No need to worry about carrying or borrowing like you do with mixed numbers. Plus, when you're dealing with negative numbers, keeping everything in fraction form helps you keep track of the signs. So, now we've transformed our problem into something much more manageable: squaring $-\frac{5}{3}$ instead of squaring $-1 \frac{2}{3}$. See how much simpler that looks? Remember, math is all about breaking down complex problems into smaller, easier-to-handle steps. And this conversion is a perfect example of that.

Squaring the Improper Fraction

Now for the fun part: squaring the improper fraction. We have $\left(-\frac{5}{3}\right)^2$, which means we need to multiply $-\frac{5}{3}$ by itself: $-\frac{5}{3} \times -\frac{5}{3}$. When multiplying fractions, we multiply the numerators together and the denominators together. So, 5 * 5 = 25 and 3 * 3 = 9. That gives us $\frac{25}{9}$. But wait, there's one more thing! We have a negative times a negative. Remember the rules of multiplication with negative numbers? A negative times a negative equals a positive! So, our answer is positive $\frac{25}{9}$. This is a classic example of how the rules of signs can change everything. If we forgot that a negative times a negative is a positive, we'd end up with the wrong answer. The beauty of fractions is that they make this multiplication process straightforward. We don't have to worry about place values or carrying over like we do with decimal multiplication. It's just numerators times numerators, denominators times denominators, and then apply the sign rule. This simplicity is why converting to improper fractions is so powerful. It turns what looks like a tricky problem into a series of easy steps. And remember, squaring a fraction is no different than squaring a whole number – you're just multiplying it by itself.

Converting Back to a Mixed Number (Optional)

We've got our answer as an improper fraction: $\frac{25}{9}$. But sometimes, it's helpful to convert it back to a mixed number so we can better understand the size of the number. To do this, we see how many times the denominator (9) goes into the numerator (25). 9 goes into 25 two times (2 * 9 = 18), with a remainder of 7 (25 - 18 = 7). So, $\frac{25}{9}$ is the same as $2 \frac{7}{9}$. This step is optional, but it can give you a better feel for the magnitude of the answer. An improper fraction like $\frac{25}{9}$ can sometimes be hard to visualize. But when we see it as $2 \frac{7}{9}$, we immediately know it's a little more than 2. This kind of number sense is really valuable in math. It helps you estimate answers, check your work, and understand the relationships between numbers. Plus, in some contexts, a mixed number might be the preferred way to express the answer. For example, if you were measuring ingredients for a recipe, you'd probably say you need $2 \frac{7}{9}$ cups of flour rather than $\frac{25}{9}$ cups. So, while converting back to a mixed number isn't always necessary, it's a good skill to have in your mathematical toolkit.

Final Answer

Alright guys, the final answer is $2 \frac{7}{9}$. We took a potentially tricky problem, broke it down into manageable steps, and conquered it! Remember, math isn't about memorizing formulas; it's about understanding the process. We converted a mixed number to an improper fraction, squared the fraction, and then converted back to a mixed number. Each step is a valuable skill in itself. By understanding these steps, you'll be well-equipped to tackle similar problems in the future. And that's what it's all about – building a strong foundation so you can confidently approach any math challenge that comes your way. So, next time you see an expression like $\left(-1 \frac{2}{3}\right)^2$, don't sweat it! Just remember the steps we've covered, and you'll be golden. Keep practicing, keep exploring, and most importantly, keep enjoying the journey of learning math! You've got this!