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

#evaluating Mixed Number Squares

Guys, let's dive into the world of mixed numbers and squares! When we're faced with an expression like $(-3 \frac{2}{3})^2$, it might seem a bit intimidating at first, but don't worry, it's totally manageable. The key is to break it down step by step and understand the underlying principles. This comprehensive guide will walk you through the process, ensuring you not only get the right answer but also grasp the concepts behind it. So, grab your pencils, and let's get started!

First, we need to convert the mixed number into an improper fraction. Remember, a mixed number is a combination of a whole number and a fraction. To convert it, we multiply the whole number by the denominator of the fraction and then add the numerator. This result becomes the new numerator, and we keep the same denominator. In our case, we have $-3 \frac{2}{3}$. So, we multiply 3 (the whole number) by 3 (the denominator), which gives us 9. Then, we add 2 (the numerator), resulting in 11. Therefore, $-3 \frac{2}{3}$ becomes $-\frac{11}{3}$. Don't forget to keep the negative sign! This is super important because the sign will affect our final answer, especially when we're dealing with squares. A negative number squared becomes positive, so keep that in mind.

Now that we have our improper fraction, $-\frac{11}{3}$, we need to square it. Squaring a number simply means multiplying it by itself. So, $(-\frac{11}{3})^2$ is the same as $(-\frac{11}{3}) \times (-\frac{11}{3})$. When multiplying fractions, we multiply the numerators together and the denominators together. In this case, we multiply -11 by -11, which gives us 121. Remember, a negative times a negative is a positive! Then, we multiply 3 by 3, which gives us 9. So, we now have $\frac{121}{9}$. This is our answer as an improper fraction, but sometimes, it's helpful to convert it back into a mixed number to better understand its magnitude.

To convert an improper fraction back to a mixed number, we divide the numerator by the denominator. The quotient (the result of the division) becomes the whole number part of the mixed number, the remainder becomes the new numerator, and we keep the same denominator. When we divide 121 by 9, we get 13 with a remainder of 4. So, the whole number part is 13, the new numerator is 4, and the denominator remains 9. Therefore, $\frac{121}{9}$ is equivalent to $13 \frac{4}{9}$. And there you have it! We've successfully evaluated $(-3 \frac{2}{3})^2$ and found that it equals $13 \frac{4}{9}$.

Remember, the key to mastering these types of problems is practice. The more you work with mixed numbers and squares, the more comfortable you'll become with the process. Don't be afraid to make mistakes – they're part of the learning journey. Just keep practicing, and you'll be a pro in no time! Also, it's a good idea to double-check your work, especially when dealing with negative signs and fractions. A small mistake in the beginning can lead to a wrong answer at the end. So, take your time, be careful, and enjoy the process of learning!

Breaking Down Mixed Numbers

Okay, so you've probably run into mixed numbers before, but let's break it down just in case. A mixed number, like our friend $-3 \frac{2}{3}$, is basically a combo deal: it's got a whole number part (the -3) and a fraction part (the $\frac{2}{3}$). Think of it like ordering a few whole pizzas and then a slice or two extra. You've got the whole pizzas and then the extra slices, right? Same concept here!

Why do we even bother with mixed numbers? Well, they're super handy for everyday situations. Imagine you're measuring ingredients for a recipe. You might need $2 \frac{1}{2}$ cups of flour. That's way easier to visualize than saying you need $\frac{5}{2}$ cups, even though they're the same amount. Mixed numbers give us a more intuitive sense of quantity in many cases. Now, when we're doing math, especially when multiplying or dividing, improper fractions often make things smoother. That's why we convert mixed numbers into improper fractions, like we did earlier.

So, let's recap the conversion process. To turn a mixed number into an improper fraction, you've got a little formula: (Whole number \times Denominator) + Numerator, and that result becomes your new numerator. The denominator stays the same. For $-3 \frac{2}{3}$, it looked like this: (-3 \times 3) + 2 = -9 + 2 = -7. Oops! We made a little mistake in our earlier calculation. See? It's always good to double-check! So, the improper fraction should be $-\frac{11}{3}$. We keep the negative sign because the original mixed number was negative. This negative sign is crucial, guys! It's like the secret ingredient that can totally change the flavor of your math dish. If you forget it, you might end up with a completely different answer.

Now, why is this conversion so important when we're squaring things? Well, when you square a mixed number directly, it can get messy. You'd have to think about squaring the whole number part, squaring the fraction part, and then dealing with the cross-terms. It's a recipe for confusion! Converting to an improper fraction first simplifies the process immensely. You just have one fraction to square, which is much more straightforward. Trust me, it's like choosing the express lane at the grocery store – way faster and less stressful.

So, remember, mixed numbers are useful for everyday life, but improper fractions are our friends when we're doing math operations like squaring. Master the art of converting between the two, and you'll be well on your way to conquering all sorts of math challenges!

Squaring Improper Fractions

Alright, now that we've got our improper fraction in tip-top shape, let's talk about squaring it. Squaring, in math-speak, simply means multiplying a number by itself. So, when we see $(-\frac{11}{3})^2$, we're really saying, "Hey, let's multiply $-\frac{11}{3}$ by $-\frac{11}{3}$!" It's like looking in a mirror – you're seeing the same thing twice.

Now, why are we focusing on squaring improper fractions? Well, remember how we converted our mixed number to an improper fraction earlier? That's because squaring a fraction is way easier than squaring a mixed number directly. We avoid a lot of potential headaches and messy calculations by sticking with fractions. It's all about making our lives easier, right? Think of it as choosing the right tool for the job. You wouldn't use a hammer to screw in a screw, would you? Same principle here!

So, how do we actually square a fraction? It's surprisingly simple. You just multiply the numerator (the top number) by itself, and then you multiply the denominator (the bottom number) by itself. That's it! It's like having two separate multiplication problems happening at the same time. For our example, $(-\frac{11}{3})^2$, we multiply -11 by -11, which gives us 121. And then we multiply 3 by 3, which gives us 9. So, $(-\frac{11}{3})^2$ becomes $\frac{121}{9}$. See? Not so scary after all!

But here's a super important thing to remember: the sign! When you multiply two negative numbers together, you get a positive number. That's why -11 times -11 equals positive 121. If we had forgotten the negative sign, we would have ended up with a completely different answer. Signs are like the secret code in math – you've got to crack them to get the right result. So, always pay close attention to whether you're dealing with positive or negative numbers.

Another way to think about squaring is that it always results in a non-negative number. Whether you square a positive number or a negative number, you'll always end up with a positive result (or zero, if you're squaring zero). This is a fundamental concept in math, and it's worth keeping in mind. It's like a built-in safety mechanism that prevents us from getting negative results when we square things.

So, to recap, squaring an improper fraction is as easy as multiplying the numerator by itself and the denominator by itself. Just remember to pay attention to the signs, and you'll be golden! This skill is crucial for all sorts of math problems, so mastering it now will definitely pay off in the long run.

Converting Back to Mixed Numbers

Okay, we've squared our improper fraction and landed on $\frac{121}{9}$. That's a perfectly valid answer, but sometimes it's more helpful to express it as a mixed number. Think of it like this: $\frac{121}{9}$ is like having 121 slices of pizza, and each pizza has 9 slices. It's hard to immediately picture how many whole pizzas you have. Converting to a mixed number helps us visualize the quantity better. It's like translating from math-speak to everyday language.

So, why bother converting back? Well, mixed numbers often give us a clearer sense of magnitude. When we see $13 \frac{4}{9}$, we instantly know we have 13 whole somethings and a little bit extra. It's much easier to grasp than just seeing $\frac{121}{9}$. Plus, in many real-world situations, mixed numbers are the preferred way to express quantities. Imagine telling someone you need $\frac{121}{9}$ cups of sugar for a recipe – they might look at you a little funny! But saying you need $13 \frac{4}{9}$ cups makes perfect sense.

Now, how do we actually convert an improper fraction back to a mixed number? It's all about division! We divide the numerator (the top number) by the denominator (the bottom number). The quotient (the result of the division) becomes our whole number part, the remainder becomes our new numerator, and we keep the same denominator. It's like a mathematical recycling process – we're just rearranging the numbers.

Let's walk through it with our example, $\frac{121}{9}$. We divide 121 by 9. 9 goes into 121 thirteen times (13 \times 9 = 117), with a remainder of 4 (121 - 117 = 4). So, our quotient is 13, and our remainder is 4. That means our mixed number is $13 \frac{4}{9}$. The 13 is the whole number part, the 4 is the new numerator, and the 9 stays as the denominator. And just like that, we've successfully converted back to a mixed number!

Think of this conversion process as undoing the original conversion from a mixed number to an improper fraction. We're essentially going in reverse. It's like driving back home after a long trip – you're retracing your steps. And just like on a road trip, it's always a good idea to double-check your route to make sure you haven't made any wrong turns. In this case, you can double-check your conversion by converting the mixed number back to an improper fraction and making sure it matches your original fraction.

So, converting back to mixed numbers is a valuable skill, especially for understanding the magnitude of your answer and for expressing quantities in a way that makes sense in real-world situations. Master this skill, and you'll be a true math whiz!

Final Thoughts and Practice

Okay, guys, we've covered a lot of ground! We've learned how to evaluate the square of a mixed number, from converting it to an improper fraction to squaring the fraction and then converting back to a mixed number. That's quite the journey! But remember, the key to mastering any math skill is practice, practice, practice. It's like learning a musical instrument – you wouldn't expect to become a virtuoso overnight, right? Same goes for math. The more you practice, the more comfortable and confident you'll become.

So, why is this skill of evaluating mixed number squares so important? Well, it's a building block for more advanced math concepts. You'll encounter squares and mixed numbers in algebra, geometry, and even calculus. Mastering the basics now will make your life much easier down the road. It's like building a strong foundation for a house – the stronger the foundation, the sturdier the house.

Plus, understanding mixed numbers and fractions is super useful in everyday life. We use fractions and mixed numbers all the time, whether we realize it or not. From measuring ingredients in the kitchen to calculating discounts at the store, these concepts are essential for navigating the world around us. So, the time you invest in learning them now will definitely pay off in the long run.

Now, let's recap the steps we've learned: 1. Convert the mixed number to an improper fraction. 2. Square the improper fraction (multiply the numerator by itself and the denominator by itself). 3. Convert the resulting improper fraction back to a mixed number (if needed). These three steps are like the magic formula for solving these types of problems. Memorize them, practice them, and they'll become second nature.

But remember, math isn't just about memorizing formulas. It's about understanding the underlying concepts. Why do we convert to improper fractions? Why do we square the numerator and denominator separately? Why do we convert back to mixed numbers? Asking these questions will help you develop a deeper understanding of the math, which will make you a more effective problem-solver. It's like understanding the recipe, not just following the instructions.

So, what's next? Grab some practice problems and start working! Don't be afraid to make mistakes – they're part of the learning process. And if you get stuck, don't hesitate to ask for help. There are tons of resources available, from textbooks and online tutorials to teachers and classmates. The key is to keep practicing and keep asking questions. With a little effort, you'll be evaluating mixed number squares like a pro in no time!

So, to finalize the calculation, $(-3 \frac{2}{3})^2 = (-\frac{11}{3})^2 = \frac{121}{9} = 13\frac{4}{9}$.

Remember to double-check your work and practice regularly. Happy calculating!