Evaluate $\frac{2ab(a+b)^2}{b-4a}$
Hey guys! Today, we're diving into a fun math problem where we need to evaluate a complex expression. Don't worry, it's not as scary as it looks! We'll break it down step by step so you can easily follow along. Our goal is to find the value of the expression $\frac{2 a b(a+b)^2}{b-4 a}$ when $a = 4$ and $b = -2$. This involves substituting the given values of a and b into the expression and then simplifying it using the order of operations. So, let's grab our calculators (or our mental math muscles!) and get started!
Before we jump into plugging in the numbers, let's take a good look at the expression we're dealing with: $\frac{2 a b(a+b)^2}{b-4 a}$. We see that it involves variables (a and b), multiplication, addition, and exponents, all wrapped up in a fraction. The key here is to remember the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). This tells us the order in which we need to perform the operations to get the correct answer. We will first substitute the values of a and b. Then, we'll tackle the operations inside the parentheses, followed by the exponent. After that, we'll handle the multiplication in the numerator and the subtraction in the denominator. Finally, we'll divide the numerator by the denominator to get our final answer. Remember, paying attention to the signs (positive and negative) is crucial in these types of problems. A small mistake with a sign can throw off the entire calculation. So, let's be careful and methodical as we work through each step. This expression is a good example of how algebraic expressions can combine different mathematical operations, making it important to have a solid understanding of the order of operations. By understanding the structure of the expression and applying the correct order of operations, we can confidently solve it. Now that we have a good grasp of what we're dealing with let's move on to the next step: substituting the values of a and b into the expression. Remember, math is like building with blocks; each step builds on the previous one, so a clear understanding of each step is essential for success!
Okay, the first step in solving this problem is to substitute the given values of a and b into our expression. We know that $a = 4$ and $b = -2$. So, everywhere we see an a, we'll replace it with 4, and everywhere we see a b, we'll replace it with -2. This gives us: $\frac{2 * 4 * (-2) * (4 + (-2))^2}{(-2) - 4 * 4}$. Notice how we've carefully replaced each variable with its corresponding value, making sure to keep the signs correct. It's a good idea to double-check this step to make sure we haven't made any mistakes, as any error here will carry through to the rest of the solution. This substitution step is a fundamental part of algebra. It allows us to take an abstract expression with variables and turn it into a concrete numerical expression that we can evaluate. By substituting values, we can see how the expression behaves for different inputs and understand the relationship between the variables. Also, it's really important to use parentheses when substituting negative values, especially when they're being multiplied or raised to a power. This helps to avoid confusion with the order of operations and ensures that we get the correct sign in our calculations. Now that we've successfully substituted the values, our next step is to simplify the expression following the order of operations. We'll start with the parentheses and then move on to the exponents, multiplication, and division. So, let's move on to the next step and continue our journey to solving this problem!
Alright, now that we've substituted the values, let's tackle the parentheses. Looking at our expression, $\frac2 * 4 * (-2) * (4 + (-2))^2}{(-2) - 4 * 4}$, we see that we have something to simplify inside the parentheses in the numerator{(-2) - 4 * 4}$. We've successfully simplified the expression inside the parentheses. This step might seem small, but it's crucial for keeping our calculations organized and avoiding errors. Parentheses are like little containers that tell us which operations to perform first. They ensure that we follow the correct order of operations and get the right answer. Simplifying inside the parentheses is often the first step in evaluating complex expressions, and it's a skill that you'll use again and again in algebra and beyond. By taking our time and carefully simplifying each part of the expression, we're building a strong foundation for the rest of the problem. Now that we've handled the parentheses, let's move on to the next operation in our order of operations: exponents. We have a term with an exponent in our expression, so let's see how we can simplify that in the next step. Keep up the great work, guys! We're making excellent progress!
Okay, we've conquered the parentheses, and now it's time to tackle the exponent. Looking at our expression, $\frac2 * 4 * (-2) * (2)^2}{(-2) - 4 * 4}$, we see the term (2)^2. Remember, an exponent tells us how many times to multiply the base by itself. So, (2)^2 means 2 multiplied by itself, which is 2 * 2 = 4. Now, let's replace (2)^2 with 4 in our expression. This gives us{(-2) - 4 * 4}$. We've successfully evaluated the exponent! Exponents are a powerful tool in mathematics, allowing us to express repeated multiplication in a concise way. Understanding how to evaluate exponents is crucial for simplifying expressions and solving equations. When dealing with exponents, it's important to remember that the exponent applies only to the base directly to its left. In this case, the exponent 2 applies only to the 2 inside the parentheses, not to the entire numerator. This is why we simplified the parentheses first and then applied the exponent. Also, be mindful of negative signs when dealing with exponents. If the base is negative, the result will be positive if the exponent is even and negative if the exponent is odd. However, in our case, the base is positive, so we don't have to worry about that. Now that we've handled the exponent, we're ready to move on to the next operations in our order of operations: multiplication and division. We have several multiplications in our numerator and a multiplication in our denominator, so let's see how we can simplify those in the next step. We're getting closer and closer to the final answer, guys! Keep up the fantastic work!
Fantastic! We've simplified the parentheses and the exponent, and now it's time for multiplication and division. Remember, multiplication and division have the same precedence in the order of operations, so we perform them from left to right. Looking at our expression, $\frac2 * 4 * (-2) * 4}{(-2) - 4 * 4}$, let's start with the numerator. We have 2 * 4 * (-2) * 4. Let's multiply these numbers together step by step. First, 2 * 4 = 8. Then, 8 * (-2) = -16. Finally, -16 * 4 = -64. So, the numerator simplifies to -64. Now, let's move on to the denominator. We have (-2) - 4 * 4. According to the order of operations, we need to perform the multiplication before the subtraction. So, 4 * 4 = 16. Now we have (-2) - 16. Subtracting a positive number is the same as adding a negative number, so (-2) - 16 is the same as (-2) + (-16), which equals -18. So, the denominator simplifies to -18. Now our expression looks like this{-18}$. We've performed all the multiplications and divisions, and we're left with a fraction. To simplify this fraction, we can divide both the numerator and the denominator by their greatest common divisor. Both -64 and -18 are divisible by -2. Dividing -64 by -2 gives us 32, and dividing -18 by -2 gives us 9. So, our simplified fraction is $\frac{32}{9}$. Multiplication and division are fundamental operations in mathematics, and mastering them is crucial for success in algebra and beyond. When performing multiplication and division, it's essential to pay attention to the signs of the numbers. A positive number multiplied by a positive number gives a positive result, a negative number multiplied by a negative number gives a positive result, and a positive number multiplied by a negative number (or vice versa) gives a negative result. The same rules apply to division. We've done an amazing job simplifying the numerator and the denominator, and now we have our final answer in the form of a fraction. Let's recap our steps and celebrate our accomplishment in the final section!
Woohoo! We've made it to the final step! After carefully substituting, simplifying, and calculating, we've arrived at our final answer. Our original expression, $\frac{2 a b(a+b)^2}{b-4 a}$, where $a = 4$ and $b = -2$, simplifies to $\frac{32}{9}$. That's it! We've successfully evaluated the expression. Guys, you've done an incredible job following along and working through this problem. Remember, the key to solving complex math problems is to break them down into smaller, more manageable steps. By following the order of operations and carefully performing each step, we can tackle even the most challenging expressions. This problem involved several important mathematical concepts, including substitution, parentheses, exponents, multiplication, division, and simplification of fractions. By mastering these concepts, you'll be well-prepared for more advanced math topics in the future. Math is like a puzzle, and each step we take is like fitting a piece into place. When we finally solve the puzzle, it's a great feeling of accomplishment! So, congratulations on solving this one with me! Keep practicing and keep exploring the wonderful world of mathematics. There's always something new to learn and discover. And remember, math can be fun! It's all about challenging ourselves and pushing our understanding further. Until next time, keep up the fantastic work, and I'll see you in the next math adventure!
