Calculate The Value Of The Expression (8-4)^2/-2 / -12/3

Hey guys! Today, we're diving into a mathematical expression that might look a little intimidating at first glance, but trust me, it's totally manageable once we break it down step by step. We're going to tackle this expression:

$\frac{\frac{(8-4)^2}{-2}}{\frac{-12}{3}}$

Don't worry; we'll go through each part methodically, so you'll not only get the answer but also understand the process. Let's get started!

Understanding the Order of Operations

Before we even think about plugging in numbers, we need to talk about the order of operations. This is like the golden rule of math, ensuring we all arrive at the same correct answer. You might have heard of the acronym PEMDAS, which stands for:

• Parentheses
• Exponents
• Multiplication and Division (from left to right)
• Addition and Subtraction (from left to right)

Think of it as a recipe; you need to follow the instructions in the right order to bake a delicious cake, and math is no different! Ignoring PEMDAS can lead to a completely wrong result, so let’s keep it top of mind as we work through our expression. In this particular expression, we have parentheses, exponents, division, and a big fraction, which is essentially another form of division. So, we'll be focusing on these aspects of PEMDAS.

Parentheses First

The first thing we need to address according to PEMDAS is the parentheses. Inside the parentheses, we have a simple subtraction: (8 - 4). This is straightforward enough: 8 minus 4 equals 4. So, we can replace (8 - 4) with 4 in our expression. This might seem like a small step, but it's crucial for simplifying the overall expression. By dealing with the parentheses first, we're reducing the complexity and making the subsequent steps easier to manage. It's like decluttering your workspace before starting a big project; a clear space helps you think more clearly and work more efficiently. So, let's make that substitution and see how our expression looks now. It's already becoming less daunting, isn't it?

Tackling the Exponent

Next up, we have an exponent! Remember that exponents tell us how many times to multiply a number by itself. In our case, we have 4 squared, which is written as 4². This means we need to multiply 4 by itself: 4 * 4. The result is 16. So, we can replace 4² with 16 in our expression. Understanding exponents is fundamental in mathematics, as they appear in various contexts, from algebra to calculus. They provide a concise way to represent repeated multiplication and are essential for dealing with growth and decay scenarios. By evaluating the exponent, we're continuing to simplify our expression, bringing us closer to the final answer. Think of it as peeling away the layers of an onion; each step reveals a simpler form of the original problem.

Simplifying the Numerator

Now that we've handled the parentheses and the exponent, let's focus on the numerator of our main fraction. The numerator is the top part of the fraction, and in our case, it's $\frac{16}{-2}$. This is a simple division problem. We're dividing 16 by -2. Remember the rules for dividing positive and negative numbers: a positive number divided by a negative number results in a negative number. So, 16 divided by -2 is -8. This means we can replace the entire numerator, $\frac{16}{-2}$, with -8. Simplifying the numerator is a crucial step because it reduces the complexity of the overall fraction. Fractions can sometimes look intimidating, but by breaking them down into smaller parts, like the numerator and denominator, we can make them much more manageable. It's like tackling a large project by dividing it into smaller, more achievable tasks.

Simplifying the Denominator

Alright, we've conquered the numerator; now it's time to tackle the denominator! The denominator is the bottom part of our main fraction, and it's $\frac{-12}{3}$. Just like with the numerator, this is a division problem. We're dividing -12 by 3. Again, we need to remember the rules for dividing positive and negative numbers. A negative number divided by a positive number results in a negative number. So, -12 divided by 3 is -4. This means we can replace the entire denominator, $\frac{-12}{3}$, with -4. By simplifying the denominator, we're making the main fraction much easier to deal with. It's like balancing an equation; we need to simplify both sides to find the solution. Similarly, simplifying both the numerator and denominator brings us closer to the final answer of our expression.

The Final Division

We've reached the final step! Our expression has been simplified to $\frac{-8}{-4}$. This is another division problem, but this time, we're dividing a negative number by a negative number. Remember the rule: a negative number divided by a negative number results in a positive number. So, -8 divided by -4 is 2. And that's our final answer! We've successfully navigated through the expression, step by step, using the order of operations and the rules of division. It might have seemed complex at the beginning, but by breaking it down and tackling each part methodically, we arrived at the solution with confidence. This final division is the culmination of all our previous work. Each step we took, from simplifying the parentheses to evaluating the exponent and dividing the numerator and denominator, has led us to this point. It's like the final brushstroke on a painting, bringing the entire artwork to completion.

Final Answer

So, after all that work, we've found that the value of the expression $\frac{\frac{(8-4)^2}{-2}}{\frac{-12}{3}}$ is 2. Great job, guys! You've successfully tackled a complex mathematical expression by breaking it down into manageable steps and applying the order of operations. Remember, math might seem intimidating at times, but with a methodical approach and a clear understanding of the rules, you can conquer any challenge. Keep practicing, and you'll become a math whiz in no time!