Simplify (-1/5)^2 + 1/6 A Step-by-Step Solution

Hey guys! Ever feel like math problems are just a jumble of numbers and symbols? Don't worry, we've all been there. Today, we're going to break down a seemingly complex expression into something super simple. We'll be tackling the problem: (-1/5)^2 + 1/6. Sounds intimidating? Trust me, it's not! We'll go through each step, explaining the logic and the math behind it, so you'll not only get the answer but also understand why it's the answer. So, grab your calculators (or your mental math muscles) and let's dive in!

When it comes to simplifying mathematical expressions, the key is to follow the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). In our case, (-1/5)^2 + 1/6, we first need to address the exponent. What does (-1/5)^2 actually mean? It means we're multiplying -1/5 by itself: (-1/5) * (-1/5). Now, remember the rules of multiplying fractions: you multiply the numerators (the top numbers) and the denominators (the bottom numbers). So, (-1 * -1) / (5 * 5) gives us 1/25. A negative times a negative results in a positive, which is why we end up with a positive fraction. This is a crucial step, and understanding the rules of signs is essential for accurate calculations. Many errors in math come from overlooking these basic principles. So, now we've simplified the exponent part, and our expression looks a little less scary: 1/25 + 1/6. We're halfway there!

Next up, we need to add the fractions 1/25 and 1/6. But, uh oh, we can't just add them directly because they have different denominators. We need to find a common denominator, a number that both 25 and 6 divide into evenly. The least common multiple (LCM) of 25 and 6 is 150. This means we need to convert both fractions into equivalent fractions with a denominator of 150. To convert 1/25 to an equivalent fraction with a denominator of 150, we need to multiply both the numerator and the denominator by 6 (because 25 * 6 = 150). This gives us (1 * 6) / (25 * 6) = 6/150. Similarly, to convert 1/6 to an equivalent fraction with a denominator of 150, we multiply both the numerator and the denominator by 25 (because 6 * 25 = 150). This gives us (1 * 25) / (6 * 25) = 25/150. Now we have two fractions with the same denominator: 6/150 and 25/150. Finally, we can add these fractions by adding their numerators and keeping the denominator the same: 6/150 + 25/150 = (6 + 25) / 150 = 31/150. And that's our final answer! We've successfully simplified the expression. Remember, the key is breaking down the problem into smaller, manageable steps and understanding the underlying rules of arithmetic.

Let's dive deeper into the step-by-step solution of simplifying the expression (-1/5)^2 + 1/6. Sometimes, seeing each step clearly laid out can make all the difference in understanding the process. We'll break down each operation, explaining the why behind the what. This is not just about getting the right answer; it's about building a solid foundation in mathematical thinking.

Our initial expression, as we know, is (-1/5)^2 + 1/6. The first hurdle to jump is the exponent. We need to evaluate (-1/5)^2. Remember, an exponent tells us how many times to multiply the base by itself. In this case, the base is -1/5, and the exponent is 2. So, we're essentially doing (-1/5) * (-1/5). When multiplying fractions, we multiply the numerators together and the denominators together. This gives us (-1 * -1) / (5 * 5). Now, here's where the rules of signs come into play. A negative number multiplied by a negative number results in a positive number. So, -1 * -1 equals 1. And 5 * 5 equals 25. Therefore, (-1/5)^2 simplifies to 1/25. This step is fundamental. Misunderstanding exponents or sign rules can lead to major errors down the line. So, make sure you've got this down! Now our expression looks cleaner: 1/25 + 1/6. We've conquered the exponent, and we're moving on to the next operation.

The next challenge is adding the fractions 1/25 and 1/6. The golden rule of adding or subtracting fractions is that they must have the same denominator. We can't simply add the numerators when the denominators are different. It's like trying to add apples and oranges – they're not the same thing! So, we need to find a common denominator. This is a number that both 25 and 6 divide into evenly. To find the least common denominator (LCD), which is the smallest number that works, we can look for the least common multiple (LCM) of 25 and 6. One way to find the LCM is to list the multiples of each number until we find a common one. Multiples of 25 are: 25, 50, 75, 100, 125, 150, ... Multiples of 6 are: 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72, 78, 84, 90, 96, 102, 108, 114, 120, 126, 132, 138, 144, 150, ... We see that 150 is the smallest number that appears in both lists. So, the LCM of 25 and 6 is 150, and that's our common denominator! This is a critical step. Finding the correct common denominator is the key to accurately adding or subtracting fractions. Now, we need to convert both 1/25 and 1/6 into equivalent fractions with a denominator of 150.

To convert 1/25 into an equivalent fraction with a denominator of 150, we need to figure out what number we need to multiply 25 by to get 150. We know from our previous work that 25 * 6 = 150. So, we multiply both the numerator and the denominator of 1/25 by 6. This gives us (1 * 6) / (25 * 6) = 6/150. Remember, multiplying the numerator and denominator by the same number doesn't change the value of the fraction; it just changes how it looks. Similarly, to convert 1/6 into an equivalent fraction with a denominator of 150, we need to figure out what number we need to multiply 6 by to get 150. We know that 6 * 25 = 150. So, we multiply both the numerator and the denominator of 1/6 by 25. This gives us (1 * 25) / (6 * 25) = 25/150. Now we have two fractions that we can easily add: 6/150 and 25/150. Finally, we can add these fractions. To add fractions with the same denominator, we simply add the numerators and keep the denominator the same. So, 6/150 + 25/150 = (6 + 25) / 150 = 31/150. And there you have it! We've successfully added the fractions and simplified the expression. Our final answer is 31/150. This step-by-step breakdown illustrates the importance of understanding each individual operation and how they fit together to solve the problem.

Math can be tricky, and it's super easy to make mistakes if you're not careful. When simplifying expressions like (-1/5)^2 + 1/6, there are a few common pitfalls that students often stumble into. Let's talk about these mistakes and, more importantly, how to dodge them! Identifying and understanding these common errors is a fantastic way to boost your math confidence and accuracy.

One of the most frequent mistakes happens right at the beginning: dealing with the exponent. Students might forget that the exponent applies to the entire fraction, including the negative sign. So, (-1/5)^2 means (-1/5) * (-1/5), not -(1/5 * 1/5). This is a crucial distinction. If you treat it as the latter, you'll miss the fact that a negative times a negative is a positive, and you'll end up with the wrong sign in your answer. The key to avoiding this is to write out the multiplication explicitly. Instead of trying to do it in your head, write down (-1/5) * (-1/5). This visual reminder can help you keep track of the negative signs and ensure you apply the exponent correctly. Another common mistake related to exponents is forgetting the basic rules of exponents altogether. Remember, a number raised to the power of 2 (squared) means multiplying it by itself. This seems simple, but it's easy to overlook under pressure. So, always double-check that you're applying the exponent correctly and not confusing it with other operations.

Another major area where mistakes often occur is when adding fractions. As we discussed earlier, you must have a common denominator before you can add fractions. A common error is to simply add the numerators and the denominators separately, which is totally incorrect. For example, some students might mistakenly add 1/25 + 1/6 as (1+1)/(25+6) = 2/31. This is a big no-no! To avoid this, always remember the fundamental rule: find the least common denominator (LCD) first. Once you have the LCD, convert each fraction into an equivalent fraction with that denominator. Only then can you add the numerators while keeping the denominator the same. It's a multi-step process, but it's essential for getting the correct answer. Another related mistake is struggling to find the correct common denominator. If you choose a common multiple that isn't the least common multiple, you can still get the right answer, but you'll end up with a fraction that needs further simplification. To minimize extra work, it's best to find the LCD right from the start. Techniques like listing multiples or using prime factorization can help you find the LCD efficiently. And guys, if you're unsure, don't hesitate to double-check your work!

Finally, a more general mistake that can trip up anyone is simply making arithmetic errors. These can be anything from miscalculating a multiplication to forgetting a carried digit. These errors might seem small, but they can throw off your entire solution. The best way to combat arithmetic errors is to be meticulous and organized. Write neatly, show your work step-by-step, and double-check each calculation. It might take a little extra time, but it's far better to catch a mistake early on than to get the whole problem wrong. Also, practice makes perfect! The more you work with numbers, the more comfortable and confident you'll become in your calculations. Don't be afraid to use a calculator to verify your work, especially on more complex calculations. But remember, understanding the underlying concepts is even more important than just getting the right answer. By being aware of these common mistakes and taking steps to avoid them, you'll be well on your way to simplifying expressions like a pro! Keep practicing, and don't get discouraged by errors – they're just opportunities to learn and improve.

Okay, so we've conquered simplifying (-1/5)^2 + 1/6. You might be thinking,