Simplifying Math Expressions: A Step-by-Step Guide

Hey math enthusiasts! Today, we're diving into the world of simplifying mathematical expressions. Specifically, we're going to break down how to evaluate an expression like this: $\frac{8-3(-2)}{4-9(-8)}$. Don't worry if it looks a little intimidating at first; we'll go through it step by step, making sure you understand every single move. This guide is designed to be super clear, so whether you're a math whiz or just starting out, you'll be able to follow along. We will use the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), to guide us.

Understanding the Order of Operations

Before we jump into the expression, let's quickly recap the order of operations. This is the golden rule of simplifying math expressions, and it ensures we all get the same answer. PEMDAS tells us the sequence in which we should tackle different parts of an expression. First, we handle anything inside Parentheses. Then, we deal with Exponents (powers and square roots). Next, we take care of Multiplication and Division, working from left to right. Finally, we handle Addition and Subtraction, also working from left to right. This systematic approach is key to getting the right answer every time. Remember, the order is crucial! Doing things out of order can lead to a completely different (and wrong) answer. Keep PEMDAS in mind, and you'll be well on your way to mastering these types of problems. Now, let's get our hands dirty with the actual expression.

Step-by-Step Evaluation of the Expression

Alright, let's tackle our expression: $\frac{8-3(-2)}{4-9(-8)}$. We'll break it down into manageable chunks to make it super easy to follow. We're going to apply PEMDAS here to make sure we do everything right. First thing first, we have to look for any parentheses. Remember, parentheses are our first priority, according to the order of operations. In our expression, we have parentheses with multiplication inside. So, we'll start there. Let's start with the numerator, which is the top part of the fraction. We see -3(-2). Multiplying -3 by -2 gives us +6, because a negative times a negative equals a positive. So, the numerator becomes 8 + 6. Now let's handle the denominator. We see -9(-8). Multiplying -9 by -8 gives us +72. So, the denominator becomes 4 + 72. So we've simplified the multiplication within the parentheses. Now we can rewrite our fraction like this: $\frac{8 + 6}{4 + 72}$.

Simplifying the Numerator and Denominator

Now that we've taken care of the multiplication within the parentheses, we can move on to the next steps. For the numerator, we have 8 + 6. Simple addition gives us 14. So, the numerator simplifies to 14. For the denominator, we have 4 + 72. Again, simple addition gives us 76. So, the denominator simplifies to 76. Now our expression looks like this: $\frac{14}{76}$. See how much simpler it's becoming? This is the power of breaking down a complex expression into smaller, more manageable steps. We've done multiplication, addition, and now we're ready to do some division, if possible. Remember that math is all about making things clear and easier to handle.

Reducing the Fraction

Our expression now is $\frac{14}{76}$. The final step is to simplify this fraction if we can. We need to find the greatest common divisor (GCD) of the numerator (14) and the denominator (76). Let's think about this: What's the largest number that can divide both 14 and 76 evenly? Well, we know that both 14 and 76 are even numbers, so they are both divisible by 2. Dividing 14 by 2 gives us 7, and dividing 76 by 2 gives us 38. So, the simplified fraction is $\frac{7}{38}$. Is there a number larger than 1 that divides both 7 and 38? No, there isn't. So, we're done! We've successfully simplified the expression from start to finish. We went from a somewhat complex fraction to a much simpler and more manageable fraction. By understanding PEMDAS and taking things one step at a time, we were able to solve it.

Conclusion

And there you have it, folks! We've successfully evaluated the expression $\frac{8-3(-2)}{4-9(-8)}$ step by step, resulting in $\frac{7}{38}$. Remember the key takeaways: Always follow the order of operations (PEMDAS), break down complex problems into smaller, manageable steps, and simplify your fractions when possible. Math might seem intimidating at first, but with a systematic approach and a little practice, you can master these types of expressions. Keep practicing, and you'll become a pro in no time! So, keep exploring, keep questioning, and most importantly, keep having fun with math! If you're tackling more complex problems, always double-check your work, and don't hesitate to seek help when you need it. There are tons of resources available online and in your community to help you succeed. Now go forth and conquer those math problems! Remember, practice makes perfect, and the more you work at it, the better you'll become. Happy calculating!