Simplifying Math Expressions: A Step-by-Step Guide

Hey guys! Let's dive into the world of simplifying math expressions. This can seem a bit intimidating at first, but trust me, with a little practice and the right approach, you'll be acing these problems in no time. Today, we're going to break down the expression: $\left(\frac{10}{11}\right)^2-\frac{1}{11} \cdot 1 \frac{1}{12}$. We'll go through each step carefully, explaining the 'why' behind each move, so you not only get the answer but also understand the process. The key here is to break down the problem into smaller, manageable chunks. We'll start with the exponents, then handle the multiplication, and finally, wrap things up with subtraction. Sounds good? Let's get started. Remember, the more you practice, the more comfortable you'll become. So grab your pencils and let's have some fun with some cool mathematics. Learning to simplify expressions is a fundamental skill in algebra and beyond, so this is a great place to start! We'll start by taking a look at the exponent first, and then go step by step. We have a mixed fraction to deal with too, so let's convert that into an improper fraction. And then we can multiply that with the fraction. It might sound like a lot of work, but trust me, it's not that bad, you will get used to it! This is also useful for other fields like computer science, physics, chemistry etc, so this is a very good skill to acquire.

Step 1: Handling the Exponent

Alright, first things first: let's tackle that exponent. We have $\left(\frac{10}{11}\right)^2$. Remember what an exponent means? It tells us to multiply the base by itself the number of times indicated by the exponent. In this case, the base is $\frac{10}{11}$, and the exponent is 2. So, we're essentially calculating $\frac{10}{11} \cdot \frac{10}{11}$. To multiply fractions, you multiply the numerators (the top numbers) together and the denominators (the bottom numbers) together. So, $10 \cdot 10 = 100$ and $11 \cdot 11 = 121$. Therefore, $\left(\frac{10}{11}\right)^2 = \frac{100}{121}$. Now, let's substitute this back into our original expression. Our expression now looks like this: $\frac{100}{121} - \frac{1}{11} \cdot 1 \frac{1}{12}$. See? We've already simplified a part of it. Nice work, everyone! The key here is to stay organized and take it one step at a time. Do not try to rush it. We are making progress, one step at a time, we will get it! Remember to write down each step, so you can go back and check your work later if needed. It is very useful and you will get used to it, I promise! Now that we have taken care of the exponent, we are on the right track!

Important Note: Always remember the order of operations (PEMDAS/BODMAS): Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). This order is crucial for getting the correct answer.

Step 2: Dealing with the Mixed Fraction and Multiplication

Okay, next up, we have a mixed fraction and a multiplication to deal with. Let's first convert that mixed fraction, $1 \frac{1}{12}$, into an improper fraction. To do this, you multiply the whole number (1) by the denominator (12) and add the numerator (1). This gives us $(1 \cdot 12) + 1 = 13$. We keep the same denominator, so $1 \frac{1}{12} = \frac{13}{12}$. Now, let's substitute this back into our expression. We get: $\frac{100}{121} - \frac{1}{11} \cdot \frac{13}{12}$. Now, we perform the multiplication: $\frac{1}{11} \cdot \frac{13}{12}$. Multiply the numerators: $1 \cdot 13 = 13$. Multiply the denominators: $11 \cdot 12 = 132$. So, $\frac{1}{11} \cdot \frac{13}{12} = \frac{13}{132}$. Now, substitute this back into the expression: $\frac{100}{121} - \frac{13}{132}$. We are almost done, guys! See, we just had to convert the mixed fraction into improper fraction and now we have to subtract. Easy as pie, isn't it? Let's get through this and have some fun!

Helpful Tip: When multiplying fractions, always check if you can simplify before you multiply. This can make the calculations easier.

Step 3: Subtracting the Fractions

Alright, we're at the final stretch! Now we need to subtract the fractions: $\frac{100}{121} - \frac{13}{132}$. Before we can subtract, we need to find a common denominator. This is the smallest number that both 121 and 132 can divide into evenly. To find the least common denominator (LCD), you can list the multiples of each number until you find a common one. Alternatively, you can find the prime factors of each denominator and use them to determine the LCD. In this case, the LCD is 132 * 11 = 1452. Now, we convert both fractions to have a denominator of 1452. For $\frac{100}{121}$, we multiply both the numerator and denominator by 12: $\frac{100 \cdot 12}{121 \cdot 12} = \frac{1200}{1452}$. For $\frac{13}{132}$, we multiply both the numerator and denominator by 11: $\frac{13 \cdot 11}{132 \cdot 11} = \frac{143}{1452}$. Now we can subtract: $\frac{1200}{1452} - \frac{143}{1452} = \frac{1200 - 143}{1452} = \frac{1057}{1452}$. The fraction $\frac{1057}{1452}$ is already in its simplest form, as 1057 and 1452 don't have any common factors other than 1. And there you have it, we are finished! Congrats guys, you made it through this step. You have come a long way!

Conclusion: The Simplified Answer

So, the simplified form of the expression $\left(\frac{10}{11}\right)^2-\frac{1}{11} \cdot 1 \frac{1}{12}$ is $\frac{1057}{1452}$. Great job, everyone! You've successfully navigated through exponents, mixed fractions, multiplication, and subtraction. Remember to practice these steps regularly. The more you work with these types of problems, the more confident you'll become. Always break down complex problems into smaller, more manageable steps, and you'll be well on your way to math mastery! Keep in mind the order of operations and always double-check your work. You are all capable of tackling these problems, keep up the great work. If you are struggling, feel free to go over each step. Rewatch and redo all the steps, if needed. Take a break if you need it. You can do this! Remember, it's all about practice and understanding the process. Now go forth and conquer more math problems! Keep up the awesome work!