Solving For X In Exponential Equations X^16 = 64p^4 And X^10 = 8p

Hey there, math enthusiasts! Ever stumbled upon a problem that looks like it’s written in a secret code? Well, today we’re going to crack one such code together. We've got these two equations staring back at us: x^16 = 64p^4 and x^10 = 8p. Our mission? To figure out what x actually is. Sounds like a fun puzzle, right? Let’s dive in and see if we can decode this math mystery. We will discuss how to approach this problem, breaking it down step by step so it becomes crystal clear.

The Challenge: Decoding the Value of x

Alright, let's break down the challenge. We are given two equations, and our goal is to find the value of x. The options provided suggest that x is somehow related to p, so we need to find a way to connect the two equations and isolate x. This is where the fun begins! To kick things off, let's rewrite our equations to make them a little easier on the eyes. Remember, the key here is to manipulate the equations in a way that helps us eliminate variables or find a common ground. Sometimes, raising both sides of an equation to a power or taking the root can help simplify things. Other times, division can be our best friend. So, let’s roll up our sleeves and get started on this mathematical adventure. We will be using exponent rules and algebraic manipulation to solve this. Make sure you're comfortable with these concepts, as they are the bread and butter of solving such problems. Feel free to brush up on them if needed before we proceed. Now, let’s start with rewriting the equations in a more manageable form. This will lay the foundation for our next steps. Think of it as setting the stage for the grand finale – finding x!

Step-by-Step Solution: Cracking the Code

So, where do we start? We have x^16 = 64p^4 and x^10 = 8p. Hmmm… they look quite different, don’t they? But don't worry, there’s a method to this math madness. Our mission is to massage these equations until they start singing the same tune. Let’s try to make the exponents of x similar so we can easily compare them. A brilliant move here would be to try and eliminate p. Why p? Because x is what we're after, and getting rid of p will simplify our quest. Think of it like focusing on the main ingredient in a recipe and setting aside the supporting players for a moment. Now, let's look at how we can do this. One way to get rid of p is to divide the equations. But before we do that, we need to make the powers of p the same. Remember, when dividing terms with exponents, it’s much easier if the exponents are related. So, let’s manipulate these equations a little bit. We might need to raise both sides of one or both equations to some power to align the exponents of p. This is a common technique in algebra, and it's like adjusting the volume on two speakers until they're at the same level. Once we have the powers of p aligned, we can divide the equations, and poof, p will be gone! This will leave us with an equation involving only x, which is exactly what we want. So, let’s get to work and see how we can make those p exponents play nice.

Manipulating the Equations: A Play with Exponents

Let's dive deeper into manipulating these equations. We have x^16 = 64p^4 and x^10 = 8p. To make the p exponents align, we can cube the second equation. Why cube? Because cubing x^10 = 8p gives us (x¹⁰⁾3 = (8p)^3, which simplifies to x^30 = 512p^3. Now, we're getting somewhere! Notice how we now have p raised to a power in both equations. This is crucial for our next step. Think of it like preparing the ingredients for a chemical reaction. You need to have the right components in the right form to get the desired outcome. In our case, the desired outcome is the elimination of p. So, we have two new versions of our equations: x^16 = 64p^4 and x^30 = 512p^3. But wait, the powers of p are still not the same! No problem, we have a plan. We need to find a way to make those exponents match. This is where our algebraic ninja skills come in handy. We might need to raise one or both equations to another power. The goal is to find the least common multiple of the exponents of p and make both exponents equal to that. This might sound a bit complicated, but trust me, it’s just a matter of careful manipulation. Once the powers of p are the same, we can proceed with the division, and watch p disappear! So, let’s put on our thinking caps and figure out what power we need to raise each equation to. It’s like solving a mini-puzzle within the bigger puzzle, keeping us on our toes and engaged in the mathematical adventure.

Dividing Equations: Bye-Bye, p!

Okay, we've prepped our equations. Now comes the exciting part: dividing them to eliminate p. But before we jump into the division, let's make sure our equations are in tip-top shape. We have x^16 = 64p^4 and x^30 = 512p^3. To make the exponents of p the same, we could manipulate these further, but let’s try a slightly different approach. Instead of making the p exponents exactly the same, let’s express p in terms of x from the second equation and substitute it into the first. This can sometimes be a more direct route to the solution. From x^10 = 8p, we can write p = x^10 / 8. Now, we have an expression for p solely in terms of x. This is like finding a secret key that unlocks the next level of the puzzle. Next, we're going to take this expression for p and plug it into the first equation, x^16 = 64p^4. This substitution will be a game-changer, as it will get rid of p completely from the equation, leaving us with an equation that only involves x. Once we have this, solving for x will be a piece of cake. This technique is a powerful one in algebra, and it's like a masterstroke in a chess game, setting us up for a checkmate. So, let’s substitute and see what happens. It’s always thrilling to see how these algebraic manipulations unfold, bringing us closer and closer to our solution.

Solving for x: The Final Showdown

Alright, we've substituted p and now we have an equation solely in terms of x. Let's see how it looks. Substituting p = x^10 / 8 into x^16 = 64p^4, we get x^16 = 64(x^10 / 8)^4. Now, it might look a bit intimidating, but don't worry, we’re going to simplify this step-by-step. This is where our algebraic skills really shine. First, let's simplify the right side of the equation. We have 64(x^10 / 8)^4. Remember, when raising a fraction to a power, we raise both the numerator and the denominator to that power. So, (x^10 / 8)^4 becomes (x¹⁰⁾4 / 8^4. And what is 8^4? It’s 4096. So, now we have x^16 = 64(x^40 / 4096). We're getting closer! Now, let’s simplify further. We can simplify the fraction 64/4096. Both numbers are divisible by 64, and 64/4096 simplifies to 1/64. So, our equation now looks like this: x^16 = x^40 / 64. We’re on the home stretch now! To solve for x, we need to get all the x terms on one side. Let’s multiply both sides by 64 to get rid of the fraction. This gives us 64x^16 = x^40. Now, we can divide both sides by x^16 (assuming x is not zero) to simplify further. Remember, dividing terms with exponents involves subtracting the exponents. This will give us an equation that’s much easier to handle. So, let’s do the division and see what we get. It’s like the final lap in a race, where we can see the finish line and we just need to push through to the end.

The Final Answer: Unveiling x

We've done the algebraic tango, and now we're ready for the grand reveal! After dividing both sides of 64x^16 = x^40 by x^16, we get 64 = x^24. This is much simpler, isn't it? Now, to find x, we need to take the 24th root of 64. Whoa, 24th root? That sounds intimidating! But don't worry, we can break it down. Remember, taking a root is the same as raising to a fractional power. So, we're looking for x = 64^(1/24). Now, let’s simplify 64. We know that 64 is 2^6. So, we have x = (2⁶⁾(1/24). Using the power of a power rule, we multiply the exponents, giving us x = 2^(6/24). And 6/24 simplifies to 1/4. So, x = 2^(1/4). We're almost there! Now, we need to see if this matches any of the options given. The options involve p, so we might need to go back and use our expression for p in terms of x to make a final comparison. But first, let’s simplify 2^(1/4). This can be written as the fourth root of 2, or √[4]2. Now, let's think about how this relates to our options. We need to express this in terms of p. Let's go back to the equation x^10 = 8p. We have x = 2^(1/4), so x^10 = (2^(1/4))10 = 2^(10/4) = 2^(5/2). So, 2^(5/2) = 8p. Now, we can write p = 2^(5/2) / 8. Since 8 is 2^3, we have p = 2^(5/2) / 2^3 = 2^(5/2 - 3) = 2^(-1/2). So, p = 1 / √2. Now, let’s look at our options and see which one matches our value of x. This final step is like connecting the dots to see the full picture, and it's so satisfying when everything clicks into place. So, let’s put it all together and find the answer!

After all that algebraic acrobatics, we found that x = 2^(1/4) and p = 1/√2. Now, let's go through the options and see which one matches our x. We have:

(A) 1/p = √2 (B) ∛(2p) = ∛(2 * 1/√2) = ∛√2 = 2^(1/6) (C) √(2p) = √(2 * 1/√2) = √√2 = 2^(1/4) (D) 1/√(2p) = 1/√(2 * 1/√2) = 1/∛√2 = 1/2^(1/4)

Aha! Option (C) matches our value of x = 2^(1/4). So, the correct answer is (C) √(2p). Phew! We did it! We cracked the code and found the value of x. This was quite a mathematical journey, but we navigated through it step-by-step, using our algebraic skills and problem-solving strategies. Give yourself a pat on the back for sticking with it and unraveling this mystery!

Key Takeaways: Lessons Learned

What a ride! We’ve not only solved for x but also learned some valuable problem-solving techniques along the way. This journey through exponents and equations has given us some key takeaways that we can use in future math adventures. First, manipulating equations is an art. We saw how raising equations to powers, dividing them, and substituting expressions can simplify complex problems. It’s like having a toolbox full of algebraic tricks, and knowing when to use which one is crucial. Another key takeaway is the power of simplification. We broke down complex expressions into smaller, more manageable parts. Remember, simplification is your friend! It can turn a scary-looking equation into a friendly one. We also saw the importance of connecting the dots. We didn’t just solve for x and stop there. We went back and related it to p to match the options given. This shows the importance of understanding the whole problem and how different parts connect. And finally, remember that practice makes perfect. The more you solve these kinds of problems, the better you’ll get at recognizing patterns and applying the right techniques. So, keep practicing and keep exploring the wonderful world of mathematics! With every problem you solve, you’re not just finding an answer, you’re building your skills and confidence. So, keep challenging yourself and enjoy the journey of learning!

Practice Makes Perfect: Similar Problems to Try

Now that we’ve conquered this problem, let's keep the momentum going! The best way to solidify your understanding is to practice similar problems. Here are a few problems that have a similar flavor to the one we just solved. Try tackling these on your own, using the techniques we discussed. 1. If a^8 = 25b^2 and a^4 = 5b, find a. 2. Given m^12 = 81n^4 and m^6 = 9n, what is m? 3. Solve for y if y^9 = 27z^3 and y^6 = 9z. Remember, the key is to manipulate the equations to eliminate one variable, solve for the other, and then substitute back if needed. Think about raising equations to powers, dividing them, or substituting expressions. Don’t be afraid to experiment and try different approaches. And most importantly, don’t get discouraged if you don’t get it right away. Math is a journey, and every mistake is a learning opportunity. So, grab a pencil and paper, and dive into these problems. The more you practice, the more comfortable you’ll become with these techniques, and the better you’ll get at solving these kinds of puzzles. And who knows, you might even start to enjoy the challenge! So, happy problem-solving, and may the math be with you!

Understanding the Problem

We are given two equations x^16 = 64p^4 and x^10 = 8p. Our task is to determine the value of x. This problem involves manipulating exponential equations and requires a solid understanding of exponent rules and algebraic manipulation. Let's break it down step by step to make it clear and easy to follow.

Method 1: Equating Powers of x

One approach is to manipulate both equations to express them in terms of x and p. We aim to eliminate p by making the powers of p equal or by expressing p in terms of x and then substituting. First, let's consider the second equation x^10 = 8p. We can express p in terms of x as follows:

p = x^10 / 8

Now, substitute this expression for p into the first equation x^16 = 64p^4: x^16 = 64 (x^10 / 8)^4

Next, simplify the equation:

x^16 = 64 (x^40 / 8^4) x^16 = 64 (x^40 / 4096)

Further simplification:

x^16 = x^40 / 64

Now, multiply both sides by 64 to get rid of the fraction:

64x^16 = x^40

Divide both sides by x^16 (assuming x ≠ 0):

64 = x^24

To find x, take the 24th root of both sides:

x = 64^(1/24)

Simplify 64 as 2^6:

x = (2⁶⁾(1/24)

Apply the power rule (a^m)n = a^(m*n):

x = 2^(6/24) x = 2^(1/4)

Thus, x is the fourth root of 2. Now we need to relate this back to p to match one of the answer choices.

Relating x Back to p

From the equation x^10 = 8p, substitute x = 2^(1/4):

(2^(1/4))10 = 8p 2^(10/4) = 8p 2^(5/2) = 8p

Now, divide by 8 (which is 2^3):

p = 2^(5/2) / 2^3 p = 2^(5/2 - 3) p = 2^(5/2 - 6/2) p = 2^(-1/2) p = 1 / √2

Checking the Answer Choices

Now, let’s check the answer choices:

(A) 1/p = √2 (B) ∛(2p) = ∛(2 * 1/√2) = ∛√2 = 2^(1/6) (C) √(2p) = √(2 * 1/√2) = √√2 = 2^(1/4) (D) 1/√(2p) = 1/√(2 * 1/√2) = 1/∛√2 = 1/2^(1/4)

The correct answer is (C) √(2p), as it matches our calculated value of x = 2^(1/4).

Method 2: Raising Equations to Powers

Another method is to raise both equations to appropriate powers to equate either x or p terms. Let’s raise the second equation to the 4th power:

(x¹⁰⁾4 = (8p)^4 x^40 = 8^4 p^4 x^40 = 4096 p^4

Now, multiply the first equation by 64:

64x^16 = 64 * 64p^4 64x^16 = 4096 p^4

Now we have two equations with 4096p^4:

x^40 = 4096 p^4 64x^16 = 4096 p^4

Equate the two expressions:

x^40 = 64x^16

Divide by x^16:

x^24 = 64

This leads us to the same solution as before:

x = 64^(1/24) = 2^(1/4)

Continue as in Method 1 to relate x back to p and check the answer choices.

Conclusion

Both methods lead us to the same answer. The value of x is 2^(1/4), which corresponds to answer choice (C) √(2p). This problem highlights the importance of algebraic manipulation and understanding exponent rules to solve exponential equations effectively.