Solving For X + Y + Z: Math Problem Breakdown

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Solving for x + y + z: Math Problem Breakdown

Hey math enthusiasts! Let's dive into a fun problem. We're given the equation 180 = 2ˣ * y * 5ᶻ and we're tasked with finding the value of x + y + z. Don't worry, it's not as scary as it looks. We'll break it down step-by-step, making it super easy to understand. This is a classic example of prime factorization and applying some basic exponent rules. Let's get started!

Understanding the Problem: Prime Factorization

Alright, guys, before we jump into the nitty-gritty, let's make sure we're all on the same page. The core concept here is prime factorization. Remember those prime numbers from elementary school? Numbers like 2, 3, 5, 7, and so on, that are only divisible by 1 and themselves. Prime factorization is all about breaking down a number into a product of its prime factors. Think of it like this: every whole number (except 1) can be written as a unique combination of prime numbers multiplied together.

So, when we see 180 = 2ˣ * y * 5ᶻ, the equation is telling us that 180 can be expressed as a product of powers of prime numbers (2 and 5) and another factor 'y'. The variables x and z are simply the exponents of 2 and 5 respectively. Our mission? To figure out what those exponents are, what 'y' is, and then add them all up. Understanding prime factorization is key to solving this type of problem. It's the foundation upon which we'll build our solution. It's also a fundamental skill in number theory, so mastering it is definitely a win-win!

Now, let's apply this knowledge to our equation. We will break down the number 180 into its prime factors. This will help us identify the values of x, y, and z. The prime factorization of 180 will give us the base for solving the rest of the problem. Don't worry, this part is pretty straightforward and a good way to refresh your math skills. Just follow along and you'll be a prime factorization pro in no time! Remember, we're aiming to express 180 as a product of prime numbers. This will allow us to compare the prime factors with the terms on the right side of the equation, thus allowing us to calculate the unknown values. Ready? Let's get cracking!

Step-by-Step Prime Factorization of 180

Let's get down to business and find the prime factors of 180, shall we? Here's how we can do it. You can follow these steps, and it works for any number. This is one of the most important concepts when working with equations like this, so pay close attention.

  1. Start Dividing: Begin by dividing 180 by the smallest prime number, which is 2. 180 / 2 = 90. So, we know that 2 is a factor of 180. We can write this as 180 = 2 * 90.
  2. Continue Dividing: Now, let's look at 90. It's also divisible by 2. 90 / 2 = 45. We now have 180 = 2 * 2 * 45.
  3. Next Prime: 45 isn't divisible by 2. So, we move to the next prime number, which is 3. 45 / 3 = 15. Our equation becomes 180 = 2 * 2 * 3 * 15.
  4. Keep Going: 15 is also divisible by 3. 15 / 3 = 5. Now, we have 180 = 2 * 2 * 3 * 3 * 5.
  5. Final Factor: Finally, 5 is a prime number, so we can't divide it further. We've reached the end! The prime factorization of 180 is 2 * 2 * 3 * 3 * 5.

Great job, guys! We've successfully broken down 180 into its prime factors. Now let's use this information to solve our original equation. By expressing 180 in terms of its prime factors, we can easily compare it to the right-hand side of our original equation (180 = 2ˣ * y * 5ᶻ) and find the values of x, y, and z. This will bring us closer to the final solution! We are on the right track! Just a few more steps!

Matching Prime Factors to the Equation

Okay, we've got the prime factorization of 180, which is 2 * 2 * 3 * 3 * 5. Now, let's match these factors to the right side of our equation, 2ˣ * y * 5ᶻ. Remember, our goal is to find the values of x, y, and z.

  1. Identify x: We have two 2's in the prime factorization of 180. The term 2Ë£ in our equation tells us that 2 is raised to the power of x. Since we have two 2's, it means x = 2. So, we've found our first value!
  2. Identify z: We have one 5 in the prime factorization of 180. The term 5ᶻ in our equation tells us that 5 is raised to the power of z. Since we have one 5, it means z = 1. Awesome, we're making progress!
  3. Identify y: Now, let's figure out y. The prime factorization of 180 is 2 * 2 * 3 * 3 * 5, and we've already accounted for the two 2's (2ˣ) and the one 5 (5ᶻ). This means that the remaining factors, which are 3 * 3, must be equal to y. So, y = 3 * 3 = 9. Sweet, we've got all the values!

See? It's all about matching the prime factors on both sides of the equation. We've carefully compared the prime factors of 180 with the terms in the equation to deduce the values of x, y, and z. This approach ensures that we don't miss any factors and correctly assign them to the corresponding variables. This is a crucial step, so take your time and make sure everything aligns properly. Once you get the hang of it, these problems become quite enjoyable!

Putting It All Together

Alright, we've done all the hard work! We've found the values of x, y, and z. Now it's time to put it all together. From our prime factorization and matching, we have:

  • x = 2
  • y = 9
  • z = 1

Remember, our original question was to find the value of x + y + z. Now that we know the values of each variable, we just need to add them. It's the final stretch, and we are almost done!

So, x + y + z = 2 + 9 + 1 = 12.

Therefore, the value of x + y + z is 12! We did it, guys!

Conclusion: Wrapping Up the Problem

And there you have it! We've successfully solved the equation 180 = 2ˣ * y * 5ᶻ and found that x + y + z = 12. We started by understanding the concept of prime factorization, then we broke down 180 into its prime factors. Next, we matched those factors to the terms in the equation to determine the values of x, y, and z. Finally, we added those values together to get our answer.

This problem highlights the importance of prime factorization and how it can be applied to solve equations. It's a fundamental concept in mathematics that has various applications. By practicing problems like this, you're not just improving your math skills, but you're also developing your problem-solving abilities.

Hopefully, you found this explanation helpful. If you have any questions or want to try some more practice problems, feel free to ask! Keep practicing, keep learning, and keep enjoying the world of math. You got this!