Solving $|3-10|-(12 ÷ 4+2)^2$: A Step-by-Step Guide

Hey guys! Let's break down this math problem together. It looks a bit intimidating at first, but trust me, it's totally manageable when we take it one step at a time. We're dealing with an expression that involves absolute values, division, addition, and exponents. Don't worry; we'll go through each operation nice and slow so you can follow along easily. By the end of this guide, you'll not only know how to solve this specific problem but also feel more confident tackling similar math questions. So, grab your pencil and paper, and let's dive in!

Understanding the Expression

Okay, let's start by understanding what we're actually looking at. The expression is $|3-10|-(12 ÷ 4+2)^2$. There are a few key parts here:

• Absolute Value: Represented by $|3-10|$, this means we need to find the magnitude (or distance from zero) of the result of $3-10$.
• Division: We have $12 ÷ 4$, which is a straightforward division operation.
• Addition: Inside the parentheses, we also have an addition operation, $(12 ÷ 4) + 2$.
• Exponent: The entire expression inside the parentheses is raised to the power of 2, indicated by the $^2$.
• Subtraction: Finally, we have a subtraction operation that combines the results from the absolute value and the exponent.

Now that we've identified all the pieces, let's start solving it step by step!

Step 1: Solving the Absolute Value

The absolute value part of our expression is $|3-10|$. Absolute value, denoted by two vertical bars, essentially asks for the distance of a number from zero. It always results in a non-negative value. So, the first thing we need to do is to solve the expression inside the absolute value bars, which is $3 - 10$. When you subtract 10 from 3, you get -7. So we have $|-7|$. The absolute value of -7 is simply 7 because -7 is 7 units away from zero. Therefore, $|3-10| = |-7| = 7$. This simplifies the first part of our original expression, making it much easier to work with moving forward.

Remember, absolute value always returns the positive equivalent of the number inside the bars. This is a crucial concept to grasp because it affects how you handle negative numbers within these expressions. Understanding absolute value is not just about memorizing a rule; it's about understanding the fundamental concept of distance on the number line. Practice with various numbers, both positive and negative, to solidify your understanding.

Also, keep in mind that absolute value bars act as grouping symbols, similar to parentheses. This means you should always evaluate what's inside them before applying any operations outside of the bars. This order of operations is crucial for getting the correct answer.

Step 2: Solving the Parentheses

Next, let's tackle what's inside the parentheses: $(12 ÷ 4 + 2)$. According to the order of operations (PEMDAS/BODMAS), we need to perform the division before the addition. So, let's start with $12 ÷ 4$. When you divide 12 by 4, you get 3. Therefore, the expression inside the parentheses becomes $(3 + 2)$. Now, we simply add 3 and 2, which equals 5. So, $(12 ÷ 4 + 2) = (3 + 2) = 5$. Now we know that the value inside the parenthesis is 5, it makes things a lot easier when we proceed to the next steps.

Always remember the order of operations! It's the golden rule that keeps everything in the right order. Parentheses (or brackets) come first, then exponents, then multiplication and division (from left to right), and finally addition and subtraction (from left to right). Skipping or misinterpreting this order can lead to drastically incorrect results, even if you know all the individual operations correctly.

Another thing to remember is that complex expressions might have nested parentheses. In such cases, always start with the innermost parentheses and work your way outwards. It's like peeling an onion layer by layer. By following this systematic approach, you can break down even the most intimidating-looking expressions into manageable chunks. Always double-check your calculations within the parentheses to avoid errors, as any mistake here will propagate through the rest of the problem.

Step 3: Applying the Exponent

Now that we've simplified the expression inside the parentheses to 5, we need to apply the exponent. Our expression now looks like this: $7 - (5)^2$. The exponent tells us to raise 5 to the power of 2, which means we need to multiply 5 by itself. So, $5^2 = 5 * 5 = 25$. This step is pretty straightforward, but it's important to get it right because it significantly impacts the final result.

When dealing with exponents, remember that they indicate repeated multiplication. For example, $5^3$ would mean $5 * 5 * 5$. It's a common mistake to confuse exponents with simple multiplication (e.g., thinking $5^2$ is $5 * 2$). Always remember that the exponent tells you how many times to multiply the base by itself.

Also, be careful with negative numbers and exponents. If you have a negative number inside parentheses raised to an exponent, the sign of the result depends on whether the exponent is even or odd. For example, $(-2)^2 = 4$ (positive because the exponent is even), but $(-2)^3 = -8$ (negative because the exponent is odd). In our case, we have a positive number, so we don't need to worry about the sign.

Make sure you understand the notation and what the exponent is asking you to do. Practice with different bases and exponents to build your confidence. This will help you avoid common errors and solve more complex expressions with ease.

Step 4: Performing the Subtraction

Okay, we're almost there! Our expression now looks like this: $7 - 25$. This is a simple subtraction problem. When you subtract 25 from 7, you get -18. So, $7 - 25 = -18$. And that's our final answer!

Subtraction might seem simple, but it's essential to pay attention to the order and signs. Subtracting a larger number from a smaller number will always result in a negative number. In our case, since 25 is larger than 7, the result is negative.

When dealing with more complex expressions, it's helpful to rewrite the subtraction as addition of a negative number. For example, $7 - 25$ can be rewritten as $7 + (-25)$. This can sometimes make it easier to visualize and avoid errors, especially when dealing with multiple negative numbers.

Also, remember that subtraction is the inverse operation of addition. This means that if $a - b = c$, then $c + b = a$. You can use this relationship to check your work. For example, since $7 - 25 = -18$, we can check that $-18 + 25 = 7$.

Final Answer

So, after breaking it down step by step, we've found that $|3-10|-(12 ÷ 4+2)^2 = -18$.

I hope this explanation helps you understand how to solve this type of math problem. Remember, the key is to take it one step at a time, follow the order of operations, and double-check your work. With practice, you'll become more confident and comfortable tackling even more complex expressions. Keep up the great work, guys!