Solve For X: X³ - 18 = 63 Equation Explained

by Axel Sørensen 45 views

Hey guys! Today, we're diving into the world of algebra to solve a fun little equation. Our mission, should we choose to accept it, is to find the value of 'x' in the equation x³ - 18 = 63. It might look a bit intimidating at first, but trust me, we'll break it down step-by-step so it’s super easy to understand. So, grab your thinking caps, and let's get started!

Understanding the Equation x³ - 18 = 63

First off, let's make sure we're all on the same page about what this equation actually means. In the equation x³ - 18 = 63, 'x³' means 'x' multiplied by itself three times (x * x * x). The equation tells us that when we subtract 18 from x³, the result is 63. Our goal is to figure out what number, when cubed and then subtracted by 18, gives us 63. To make it even clearer, let's go through each part piece by piece. The term x cubed (x³) is the variable term we're trying to isolate. The '- 18' is a constant that's being subtracted, and the '= 63' tells us the final result of the left side of the equation. By understanding each part, we can strategically manipulate the equation to solve for 'x'. This kind of problem is fundamental in algebra, and mastering it opens doors to more complex mathematical concepts. So, let’s dive in and make sure we nail it! By the end of this, you'll feel like a total algebra whiz, I promise! Remember, math is like building blocks, so understanding these basics is super crucial for the future. Okay, enough pep talk, let’s get down to the nitty-gritty and solve this thing!

Step-by-Step Solution to Finding x

Alright, let’s get to the fun part: solving for 'x'! To find the value of 'x', we need to isolate it on one side of the equation. This means we want to get 'x³' all by itself first. Looking at our equation, x³ - 18 = 63, we see that we have a '- 18' hanging out on the same side as 'x³'. To get rid of it, we need to do the opposite operation. Since 18 is being subtracted, we're going to add 18 to both sides of the equation. Remember, in algebra, what we do to one side, we gotta do to the other to keep things balanced! So, let’s add 18 to both sides:

x³ - 18 + 18 = 63 + 18

This simplifies to:

x³ = 81

Awesome! We've got 'x³' by itself. Now, here comes the tricky part. We need to figure out what number, when multiplied by itself three times, equals 81. In mathematical terms, we need to find the cube root of 81. This is the number that, when cubed, gives us 81. To find the cube root, we can think about perfect cubes – numbers that are the result of cubing an integer. For example, 2 cubed (2³) is 8, 3 cubed (3³) is 27, and so on. We’re looking for a number that, when cubed, equals 81. Let's try a few numbers. Is it 3? Well, 3 * 3 * 3 = 27, so that's not it. Is it 4? 4 * 4 * 4 = 64, getting closer! How about 5? 5 * 5 * 5 = 125, which is too big. Hmmm, this is a bit of a head-scratcher. It seems like 81 isn’t a perfect cube of an integer. This means our answer for 'x' might not be a whole number, and we might need to use a calculator or some more advanced methods to find the exact cube root of 81. But for now, let’s circle back to the multiple-choice options provided and see if one of them fits. Sometimes, the answer is right there in front of us! We've done the hard part of simplifying the equation, now let’s see if we can use that knowledge to pick the right answer.

Evaluating the Multiple-Choice Options

Okay, so we've simplified the equation to x³ = 81. Now, let's put on our detective hats and check the multiple-choice options. We need to figure out which of the given values for 'x', when cubed, equals 81. Remember, cubing a number means multiplying it by itself three times. Let's go through each option one by one:

  • A) 3: If x = 3, then x³ = 3 * 3 * 3 = 27. This is definitely not equal to 81, so option A is out.
  • B) 4: If x = 4, then x³ = 4 * 4 * 4 = 64. Nope, 64 is not 81, so we can cross out option B.
  • C) 5: If x = 5, then x³ = 5 * 5 * 5 = 125. This is way bigger than 81, so option C is not the right answer.
  • D) 6: We don't have option D) 6 in the question, but we need to solve the equation x³ = 81. Let's take the cube root of 81 to find the value of x. The cube root of 81 is approximately 4.3267.

It seems like there might be a slight issue with the options provided, or perhaps the question intended for a different set of choices. The closest whole number we got to 81 when cubing was 64 (from 4³), but that's still not quite it. Since none of the given options perfectly satisfy the equation x³ = 81, it’s possible there was a typo in the original question, or the correct answer is not a whole number. In a real-world test situation, this would be a good time to double-check the question and the options to make sure everything is written correctly. If everything seems correct, and none of the options fit, you might consider selecting the closest answer or consulting with your teacher or professor. For our purposes here, we've learned the process of solving the equation, and we've seen that none of the provided options are the perfect fit. That’s still a win in terms of understanding the math! So, let’s take a step back and recap what we’ve done.

Recap and Final Thoughts

Alright, guys, let's take a moment to recap what we've accomplished today. We started with the equation x³ - 18 = 63 and our mission was to find the value of 'x'. We walked through the steps of isolating 'x³' by adding 18 to both sides, which gave us x³ = 81. Then, we dove into finding the cube root of 81, which is the number that, when multiplied by itself three times, equals 81. We tried out the multiple-choice options, cubing each one to see if it matched 81. Unfortunately, none of the options (3, 4, and 5) gave us a perfect match. We found that 3³ = 27, 4³ = 64, and 5³ = 125. This means that either there was a slight mistake in the question or the answer isn't a whole number. But hey, that's totally okay! The most important thing is that we understood the process of solving for 'x'. We learned how to manipulate the equation to isolate the variable and how to think about cube roots. Even though we didn't find a perfect answer within the given options, we still learned a ton about algebra and problem-solving. Math isn't always about finding the right answer on the first try. It’s about understanding the steps, thinking critically, and being persistent. So, give yourselves a pat on the back for tackling this problem with me! You've gained valuable skills that you can use in future math challenges. Remember, practice makes perfect, so keep solving equations and exploring the awesome world of math!

I hope you found this breakdown helpful and that you feel a little more confident in your algebra skills. If you have any other math questions or topics you'd like to explore, just let me know. Keep up the great work, and I'll catch you in the next math adventure!