Solving Equations: Finding The Value Of 64ˣ * 4ʸ
Hey guys, let's dive into a cool math problem! We're given an equation: 6x + 2y = 5. Our mission? To find the value of 64ˣ * 4ʸ. Sounds like a fun challenge, right? Don't worry, we'll break it down step by step to make it super easy to understand. We'll use a mix of algebra and some clever tricks to get to the answer. So, grab your pencils, and let's get started!
Understanding the Problem: The Core Concepts
Alright, first things first, let's make sure we're all on the same page. The problem is asking us to find the value of an expression (64ˣ * 4ʸ) when we have some information about the variables x and y (6x + 2y = 5). This is a classic algebra problem where we have to use the given equation to manipulate and simplify the expression we're trying to evaluate. The key to solving this kind of problem is often to rewrite the given expression in terms of the information we have. This usually involves using properties of exponents and a little bit of algebraic manipulation. Remember that these problems are like puzzles, and each step you take brings you closer to solving it. Let's make sure we're clear about the basics before we start diving in. The primary goal here is to manipulate the expression 64ˣ * 4ʸ so that we can substitute the known value from the equation 6x + 2y = 5. Understanding the properties of exponents will be incredibly important. For example, we know that a number raised to a power and another number raised to a power can often be combined using the properties of exponents. We need to be familiar with how to express numbers as powers of the same base. Keep in mind that understanding what the question is asking is just as important as knowing how to solve it. Before starting to solve, be sure you understand the core concepts and the goal of the problem. This will help you find the right approach to solve the problem and you will be more comfortable with the solution.
Breaking Down the Expression
Let's take a closer look at the expression 64ˣ * 4ʸ. Notice that 64 and 4 are both powers of 2. We can rewrite 64 as 2⁶ (because 2 * 2 * 2 * 2 * 2 * 2 = 64) and 4 as 2² (because 2 * 2 = 4). This is the key to simplifying the expression. When we rewrite the expression using a common base, it becomes easier to handle the exponents. By expressing everything in terms of the same base, we'll be able to use the exponent rules to our advantage. Remember, in math, rewriting things in simpler terms often makes a big difference. Think about it like simplifying a complex recipe. You can break down complex ingredients into their basic components. Now, let’s rewrite the expression. We can express 64ˣ * 4ʸ as (2⁶)ˣ * (2²)ʸ. Using the power of a power rule, which states that (am)n = a^(m*n), we simplify (2⁶)ˣ to 2^(6x) and (2²)ʸ to 2^(2y). So, now our expression is 2^(6x) * 2^(2y). With the same base, the next step becomes clear to us, we can combine the exponents.
Applying Exponent Rules
Now we've got the expression in a much more manageable form: 2^(6x) * 2^(2y). Here's where another important exponent rule comes in handy: when you multiply two numbers with the same base, you add their exponents. This rule is a lifesaver in situations like this! So, 2^(6x) * 2^(2y) simplifies to 2^(6x + 2y). Now we're getting somewhere! Remember that we were given 6x + 2y = 5? It's like the problem is practically begging us to use this information. The expression 2^(6x + 2y) looks very similar to the information provided in the question. We have successfully transformed our expression into a form where we can directly use the given equation. This is a critical step because now we can substitute the known value. Using the equation 6x + 2y = 5, we can substitute 5 for (6x + 2y) in the expression. Let's do that in the next section. We're getting closer to our final answer. Just hang tight, and don’t worry if you don’t get it at first. With practice, you'll become a pro at this. Remember to always look for ways to rewrite your expressions using exponent rules or the same base to make things easier. This is a common and important technique used in a variety of math problems.
Solving for the Final Answer
Substituting the Known Value
Okay, here's where the magic happens! We've simplified the expression 64ˣ * 4ʸ to 2^(6x + 2y). And we know that 6x + 2y = 5. So, we can directly substitute 5 for (6x + 2y) in the expression. This gives us 2⁵. Now, finding the final answer is a breeze. The expression is now simplified to 2⁵. Calculating this is pretty straightforward; just multiply 2 by itself five times (2 * 2 * 2 * 2 * 2). Doing the math, 2⁵ = 32. And there you have it! We've solved the problem. It is much easier to solve when you have a direct value to substitute in the equation. Congratulations, guys, that's the final solution! Let's just summarize the whole process to make sure everything's clear. First, we wrote 64ˣ * 4ʸ as (2⁶)ˣ * (2²)ʸ. After that, we used the power of a power rule to get 2^(6x) * 2^(2y). Then, by using the rule of multiplying exponents with the same base, we got 2^(6x + 2y). Finally, since we knew that 6x + 2y = 5, we found our answer to be 2⁵, which is 32. This approach highlights how the clever use of exponents and a simple substitution can solve the most complicated questions. Remember that practice is key to mastering these techniques.
The Final Calculation
So, we've got our simplified expression: 2⁵. Now, let's calculate the final answer. 2⁵ means 2 multiplied by itself five times: 2 * 2 * 2 * 2 * 2 = 32. Therefore, the value of 64ˣ * 4ʸ is 32. When you are solving a math problem, it's always useful to double-check your work to be sure that you haven't made any mistakes. You can go back through each step to make sure everything is correct. In this case, we can see that we have taken each step, from simplification to substitution, and followed it correctly. We started with the original equation and expression, used exponent rules to simplify the expression, and then substituted the known value to arrive at our answer. Remember, in algebra, accuracy is very important! We have calculated the final value, and we're confident that our answer is correct. So, the value is 32, which matches option (c). This final answer is not just the result of computation; it is also a testament to our understanding of the mathematical concepts and the strategic use of our problem-solving skills.
Conclusion: Wrapping Things Up
Alright, folks, we did it! We successfully found the value of 64ˣ * 4ʸ, and the answer is 32. We went through all the steps, from breaking down the problem to using exponent rules and substituting the known value. Remember, math problems are like puzzles. By using the right tools and a little bit of creativity, you can solve them. Keep practicing, and you'll get better and better at it. The more you work on these types of problems, the more familiar you’ll become with the strategies. Every problem you solve adds to your math toolkit. So, congratulations again on solving this problem! Keep up the great work, and remember to always look for the most effective approach to any problem. We learned about the properties of exponents, how to rewrite expressions with the same base, and how to use substitution to find the final answer. This problem also showcases the importance of understanding the basics. Make sure to review the key concepts we discussed, and if you have any questions, feel free to ask. Stay curious, keep learning, and keep practicing, and you'll become math wizards in no time!