Evaluating (x^6 - X) / (4y) For X=-4 And Y=4 A Step-by-Step Guide

Hey guys! Today, we're diving into a fun math problem where we need to evaluate an expression. Don't worry, it's not as scary as it sounds! We're going to break it down step by step so it's super easy to follow. Our mission is to figure out the value of the expression (x^6 - x) / (4y) when x is -4 and y is 4. So, let's grab our calculators (or our brains, if you're feeling extra smart!) and get started!

Breaking Down the Expression

Understanding the Problem

Before we jump into plugging in numbers, let's make sure we understand exactly what the problem is asking. We have an algebraic expression, which is a fancy way of saying a combination of numbers, variables (like x and y), and mathematical operations (like addition, subtraction, multiplication, and division). Our goal is to find the numerical value of this expression when we know the values of x and y. Think of it like a puzzle where we have some missing pieces (the values of x and y), and once we fill them in, we can solve for the final answer.

The key to solving this problem is following the order of operations, which you might remember as PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). This tells us the order in which we need to perform the calculations to get the correct answer. So, we'll start with any exponents, then handle multiplication and division, and finally, take care of any addition or subtraction. Remember this order, and you'll be a math whiz in no time!

Plugging in the Values

Alright, let's get to the good stuff! We're given that x = -4 and y = 4. The first step is to substitute these values into our expression, (x^6 - x) / (4y). This means we're going to replace every instance of 'x' with '-4' and every instance of 'y' with '4'. Make sure you pay close attention to the signs (positive or negative) – they matter a lot in math! So, our expression now looks like this: ((-4)^6 - (-4)) / (4 * 4). See? We've just swapped the letters for the numbers. Now, it's all about simplifying this expression to find our answer.

When we substitute the values, it's crucial to handle the negative signs carefully. Remember that a negative number raised to an even power becomes positive, while a negative number raised to an odd power stays negative. This is a common area where people make mistakes, so let's be extra careful! Also, make sure to use parentheses correctly to avoid any confusion, especially when dealing with negative numbers. This will help us keep track of our calculations and ensure we get the right answer. So far, so good, right? We're on our way to cracking this problem!

Simplifying the Expression

Now comes the fun part – simplifying! We've got our expression with the values plugged in: ((-4)^6 - (-4)) / (4 * 4). Following PEMDAS, we need to tackle the exponent first. What is (-4)^6? This means -4 multiplied by itself six times: (-4) * (-4) * (-4) * (-4) * (-4) * (-4). Since we have an even number of negative signs, the result will be positive. If you multiply it out, you'll find that (-4)^6 = 4096. So, we can replace (-4)^6 with 4096 in our expression. Our expression now looks like this: (4096 - (-4)) / (4 * 4).

Next up, we need to deal with the subtraction inside the parentheses. We have 4096 - (-4). Remember that subtracting a negative number is the same as adding its positive counterpart. So, 4096 - (-4) becomes 4096 + 4, which equals 4100. Now our expression looks even simpler: 4100 / (4 * 4). See how we're gradually making the expression easier to handle? We're almost there!

Finally, let's take care of the multiplication in the denominator. We have 4 * 4, which is simply 16. So, our expression is now super streamlined: 4100 / 16. We've done all the hard work of exponents, subtraction, and multiplication. Now, all that's left is one simple division problem. Are you ready to finish this off?

Final Calculation and Answer

Okay, we've arrived at the last step: dividing 4100 by 16. Grab your calculator (or your brainpower!) and let's do this. When you divide 4100 by 16, you get 1025/4. So, the value of our expression when x = -4 and y = 4 is 1025/4. Woohoo! We did it!

Key takeaway: Remember to follow the order of operations (PEMDAS) and be careful with those negative signs. Plugging in the values correctly and simplifying step-by-step will lead you to the right answer every time. Math can be like a cool puzzle if you approach it methodically, and we just solved this one like pros!

So, if we look back at our options:

A. -1,023/4 B. 1,025/4 C. 1,023/4 D. 16,385/4

The correct answer is B. 1,025/4. Great job, everyone! You've successfully evaluated the expression. High fives all around!

Common Mistakes to Avoid

Sign Errors

One of the most common pitfalls when evaluating expressions, especially those involving negative numbers, is making mistakes with signs. As we discussed earlier, negative numbers raised to even powers become positive, while those raised to odd powers remain negative. For example, (-2)^2 is 4 (because -2 * -2 = 4), but (-2)^3 is -8 (because -2 * -2 * -2 = -8). It's super important to keep track of these sign changes, or you might end up with the wrong answer. A simple way to avoid this is to write out each step clearly, paying close attention to whether you're multiplying or dividing by a negative number. Trust me, taking a little extra time to double-check your signs can save you a lot of headaches in the long run!

Another area where sign errors can creep in is when subtracting a negative number. Remember that subtracting a negative is the same as adding a positive. So, if you see something like 5 - (-3), it's the same as 5 + 3, which equals 8. Don't let those double negatives trip you up! Practice a few examples, and you'll become a pro at handling them. And hey, if you do make a mistake, don't sweat it – just learn from it and move on. We all make errors sometimes; it's part of the learning process.

Order of Operations

Another frequent mistake is messing up the order of operations. We talked about PEMDAS earlier, and it's a lifesaver when it comes to simplifying expressions. If you skip a step or do things in the wrong order, you're likely to end up with an incorrect answer. For example, if you have an expression like 2 + 3 * 4, you need to do the multiplication before the addition. So, 3 * 4 is 12, and then 2 + 12 is 14. If you accidentally added 2 and 3 first, you'd get 5 * 4, which is 20 – a totally different result!

To avoid order of operations errors, it helps to write out the steps clearly, just like we did in the example above. Break the expression down into smaller parts, and tackle each part according to PEMDAS. Use parentheses to group terms if needed, and double-check that you're following the correct sequence. With practice, PEMDAS will become second nature, and you'll be simplifying expressions like a math whiz!

Arithmetic Errors

Sometimes, the simplest mistakes can trip us up. Arithmetic errors, like accidentally adding or multiplying numbers incorrectly, can happen to anyone, especially when dealing with larger numbers or complex expressions. Maybe you added 25 and 37 and got 61 instead of 62, or perhaps you multiplied 12 by 8 and came up with 98 instead of 96. These little slip-ups can throw off your entire calculation, so it's worth taking a few precautions to minimize them.

One way to reduce arithmetic errors is to use a calculator, especially for more complicated calculations. Calculators are great for ensuring accuracy, and they can save you time too. But even if you're using a calculator, it's still a good idea to double-check your work. Make sure you've entered the numbers correctly and that you're pressing the right buttons. Another helpful strategy is to estimate your answer before you do the calculation. This can give you a sense of whether your final answer is reasonable. If your estimate is way off from your calculated result, it's a sign that you might have made a mistake somewhere. Remember, math is all about precision, so a little extra attention to detail can go a long way!

Practice Problems

Practice Problem 1

Evaluate the expression when a = -2 and b = 5:

(3a^3 + 4b) / (2a)

Let's walk through the steps together! First, substitute the values of a and b into the expression. This gives us:

(3 * (-2)^3 + 4 * 5) / (2 * -2)

Next, we follow the order of operations (PEMDAS). We start with the exponent: (-2)^3 = -8. Now our expression looks like this:

(3 * -8 + 4 * 5) / (2 * -2)

Now, we handle the multiplication: 3 * -8 = -24 and 4 * 5 = 20. So the expression becomes:

(-24 + 20) / (2 * -2)

Next, we do the addition in the numerator: -24 + 20 = -4. And we multiply in the denominator: 2 * -2 = -4. So our expression is now:

-4 / -4

Finally, we divide: -4 / -4 = 1. So the value of the expression is 1.

Key takeaway: Break down the problem into smaller steps, and don't rush. Double-check each step as you go, and you'll be well on your way to solving it correctly!

Practice Problem 2

Evaluate the expression when x = 3 and y = -1:

(x^4 - 2x) / (5y)

Ready to tackle this one on your own? Remember the tips we've talked about: substitute the values carefully, follow the order of operations, and watch out for those negative signs. Give it your best shot, and let's see what you come up with! You've got this!

Conclusion

So, there you have it, guys! We've walked through how to evaluate algebraic expressions by substituting values and simplifying using the order of operations. We've also highlighted common mistakes to avoid and worked through a couple of practice problems. Evaluating expressions might seem tricky at first, but with a bit of practice and attention to detail, you'll become a pro in no time! Remember, math is like building blocks – each concept builds on the one before it. So, keep practicing, keep asking questions, and most importantly, keep having fun with it! You're all doing an awesome job, and I'm super proud of your progress. Keep up the great work, and I'll catch you in the next math adventure!