Solve $(-4)(-3)^2+15 \div -5$: A Step-by-Step Guide

Hey guys! Let's break down this math problem: $(-4)(-3)^2+15 \div -5$. It looks a bit intimidating at first, but don't worry, we'll tackle it together step by step. We'll not only solve it but also understand the order of operations (PEMDAS/BODMAS) that governs how we approach such expressions. So, grab your calculators (or just your brainpower!) and let's dive in!

Understanding the Order of Operations

Before we even think about plugging in numbers, we need to talk about the order of operations. This is the golden rule in mathematics that tells us in what sequence we should perform calculations. Think of it as the grammar of math; without it, our expressions would be a jumbled mess. The most common mnemonic for this is PEMDAS, which stands for:

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

Or, you might have heard of BODMAS:

• Brackets
• Orders
• Division and Multiplication (from left to right)
• Addition and Subtraction (from left to right)

They both mean the same thing! The important thing is to remember the hierarchy: Parentheses/Brackets and Exponents/Orders come first, then we deal with Multiplication and Division (in the order they appear from left to right), and finally, Addition and Subtraction (again, from left to right).

Why is this order so important? Imagine if we didn't have it. We could interpret the same expression in multiple ways and get different answers! This would be a mathematical catastrophe. The order of operations ensures that everyone, everywhere, solves the same problem and arrives at the same correct answer.

Let's illustrate with a simple example. Consider the expression $2 + 3 * 4$. If we just went from left to right, we'd do $2 + 3$ first, getting 5, and then multiply by 4, resulting in 20. But, according to PEMDAS/BODMAS, we should multiply first: $3 * 4 = 12$, and then add 2, giving us 14. See the difference? The correct answer is 14.

Understanding this order is absolutely crucial for simplifying any mathematical expression, especially those involving multiple operations. It's the foundation upon which we'll solve our main problem, so make sure you've got it down! Think of it as your math superpower – with it, you can conquer any equation.

Breaking Down the Problem: $(-4)(-3)^2+15 ext{ div } -5$

Okay, now that we've got PEMDAS/BODMAS firmly in our minds, let's apply it to our problem: $(-4)(-3)^2+15 \div -5$. This expression has a mix of multiplication, exponents, addition, and division, so the order of operations is key.

Let's walk through it step-by-step:

1. Exponents: The first thing we need to tackle is the exponent: $(-3)^2$. Remember, this means $(-3)$ multiplied by itself: $(-3) * (-3) = 9$. So, we can rewrite our expression as $(-4)(9) + 15 \div -5$.

   • Key Point: Notice that we dealt with the exponent before anything else. This is because exponents come before multiplication, division, addition, and subtraction in the order of operations. Getting this order right is vital for a correct solution.

2. Multiplication and Division (from left to right): Now we have multiplication and division. We perform these operations in the order they appear from left to right. First up is $(-4)(9)$, which equals $-36$. Our expression now looks like this: $-36 + 15 \div -5$.

   • Left-to-Right Rule: It’s crucial to remember that when you have both multiplication and division (or both addition and subtraction), you work from left to right. This prevents ambiguity and ensures everyone solves it the same way.

3. Next, we have the division: $15 \div -5$. This gives us $-3$. So, our expression simplifies to $-36 + (-3)$.

   • Sign Awareness: Pay close attention to the signs (positive and negative) of the numbers. A negative divided by a negative is positive, a positive divided by a negative is negative, and so on. Keeping track of the signs is super important for avoiding errors.

4. Addition: Finally, we have addition: $-36 + (-3)$. Adding a negative number is the same as subtracting, so this becomes $-36 - 3$, which equals $-39$.

   • Adding Negatives: Remember the rules for adding negative numbers. If the signs are the same, you add the numbers and keep the sign. If the signs are different, you subtract the smaller number from the larger number and take the sign of the larger number.

So, after carefully following the order of operations, we've arrived at our answer: $-39$. See? It wasn't so scary after all! By breaking it down into smaller, manageable steps, we conquered the problem. Remember, the key is to stay organized, apply PEMDAS/BODMAS diligently, and pay attention to those signs.

The Solution: $(-4)(-3)^2+15 ext{ div } -5 = -39$

Alright, let's recap! We've successfully navigated the expression $(-4)(-3)^2+15 \div -5$, and we've landed on the final answer. It's always a good feeling to solve a math problem, right? But it's not just about getting the answer; it's about understanding the process we used to get there. So, let's quickly review the steps we took:

1. We started by recognizing the importance of the order of operations (PEMDAS/BODMAS). This was our guiding principle, ensuring we tackled the expression in the correct sequence.
2. First, we addressed the exponent: $(-3)^2$, which simplified to 9. This transformed our expression into $(-4)(9) + 15 \div -5$.
3. Next, we handled the multiplication and division from left to right. $(-4)(9)$ gave us $-36$, and $15 \div -5$ resulted in $-3$. Our expression was now $-36 + (-3)$.
4. Finally, we performed the addition: $-36 + (-3)$, which equals $-39$.

So, the final answer is: -39

Key Takeaway: The meticulous application of the order of operations is what allowed us to simplify this complex expression and arrive at the correct solution. Without it, we'd be lost in a sea of possibilities, potentially getting a completely different (and incorrect) answer.

It's not just about memorizing PEMDAS/BODMAS; it's about understanding why it works. It's about building a solid foundation in mathematical thinking. Once you grasp this concept, you'll be able to confidently tackle a wide range of algebraic expressions and equations.

Now, go forth and practice! The more you work with these types of problems, the more comfortable and confident you'll become. And remember, math isn't about magic; it's about logical steps and clear thinking. You've got this!

Common Mistakes to Avoid

Math problems can be tricky, and it's easy to stumble if you're not careful. When dealing with expressions like $(-4)(-3)^2+15 \div -5$, there are a few common pitfalls that students often encounter. Being aware of these mistakes can help you avoid them and boost your accuracy.

1. Ignoring the Order of Operations: This is the biggest culprit! As we've emphasized, PEMDAS/BODMAS is your best friend. If you start multiplying before addressing the exponent, or add before dividing, you're heading down the wrong path. For example, if you mistakenly calculated $(-4) * (-3)$ first, you'd throw off the entire solution.

   • How to Avoid: Always write out the steps, explicitly identifying which operation you're performing at each stage. This helps keep you organized and on track.

2. Sign Errors: Negative signs can be sneaky! Forgetting a negative sign or misapplying the rules of sign multiplication and division can lead to incorrect answers. For instance, a common mistake is to incorrectly calculate $15 \div -5$ as 3 instead of -3.

   • How to Avoid: Double-check each step involving negative numbers. Use parentheses to clearly separate negative numbers and operations. Remember the rules: negative times negative equals positive, positive times negative equals negative, and so on.

3. Misinterpreting Exponents: Exponents can be confusing if you don't fully grasp their meaning. For example, $(-3)^2$ means $(-3) * (-3)$, not $-3 * 2$. Failing to correctly evaluate the exponent will lead to a wrong result.

   • How to Avoid: Write out the multiplication explicitly. This can help you visualize what the exponent is telling you to do. Remember that the exponent only applies to the term immediately to its left, unless there are parentheses indicating otherwise.

4. Left-to-Right Confusion: When you have a mix of multiplication and division (or addition and subtraction), remember to work from left to right. Skipping this rule can lead to errors. For example, in the expression above, you need to perform $(-4)(9)$ before $15 \div -5$.

   • How to Avoid: Underline or highlight the operations as you perform them, working sequentially from left to right. This helps you maintain the correct order.

5. Rushing Through the Problem: It's tempting to speed through calculations, especially when you feel confident. However, rushing increases the likelihood of making careless errors.

   • How to Avoid: Take your time! Work methodically, showing all your steps. Check your work as you go, and don't be afraid to slow down and double-check if something feels off.

By being aware of these common mistakes and actively working to avoid them, you'll significantly improve your accuracy and confidence in solving mathematical expressions. Remember, practice makes perfect, so keep at it!

Practice Problems

Now that we've dissected the problem $(-4)(-3)^2+15 \div -5$, understood the order of operations, and identified common pitfalls, it's time to put your knowledge to the test! The best way to solidify your understanding of any math concept is through practice. So, let's tackle a few more problems that are similar in structure but with different numbers. Get your pencils and paper ready, and let's dive in!

Here are a few practice problems for you to try:

1. $(-2)(-5)^2 + 20 \div -4 = ?$
2. $(6)(-2)^3 - 18 \div (-3) = ?$
3. $-3(4)^2 + 25 \div 5 = ?$
4. $(5)(-1)^4 - 12 \div (-2) = ?$
5. $-4(-3)^3 + 16 \div (-8) = ?$

Remember to follow these steps for each problem:

1. Write down the problem clearly. This helps you avoid making mistakes due to misreading the expression.
2. Identify the order of operations (PEMDAS/BODMAS). This is your roadmap for solving the problem.
3. Perform the operations step-by-step, showing your work clearly. This makes it easier to track your progress and identify any errors.
4. Double-check your work at each step. This helps you catch mistakes early on.
5. Circle your final answer. This makes it easy to see the result.

Don't be afraid to make mistakes! Mistakes are a natural part of the learning process. When you make a mistake, take the time to understand why you made it. This will help you avoid making the same mistake in the future.

(Answers: 1. -40, 2. -42, 3. -43, 4. 11, 5. 106)

By working through these practice problems, you'll gain confidence in your ability to solve expressions involving the order of operations. Keep practicing, and you'll become a math whiz in no time!