Solving The Equation 3(x+9)^(3/4) = 24 A Step-by-Step Guide
Hey there, math enthusiasts! Today, we're diving into the fascinating world of equations and tackling a particularly interesting one: 3(x+9)^(3/4) = 24. If you're scratching your head wondering how to solve this, don't worry! We're going to break it down step-by-step, making sure you not only get the answer but also understand the process. So, buckle up and let's get started!
Deconstructing the Equation
Before we jump into the solution, let's first understand what we're dealing with. The equation 3(x+9)^(3/4) = 24 involves a variable, x, raised to a fractional exponent. These types of equations might seem intimidating at first, but with a systematic approach, they become quite manageable. Our main goal is to isolate 'x' on one side of the equation. To do this, we'll need to undo the operations that are affecting 'x', one by one.
First things first, we notice that the term (x+9)^(3/4) is being multiplied by 3. To reverse this, we'll divide both sides of the equation by 3. This gives us our new equation: (x+9)^(3/4) = 8. See? We've already made progress! Now, we're left with a term raised to a fractional exponent. This is where things get a little more interesting, but don't fret, we've got this!
Fractional exponents can be a bit tricky, but they're essentially a combination of exponents and roots. The numerator of the fraction (in this case, 3) represents the power to which we're raising the base, and the denominator (in this case, 4) represents the root we're taking. So, (x+9)^(3/4) can be thought of as the fourth root of (x+9) cubed. To undo this, we need to reverse both the root and the power. The most effective way to do this is to raise both sides of the equation to the reciprocal of the fractional exponent. In our case, the reciprocal of 3/4 is 4/3. So, we'll raise both sides of the equation to the power of 4/3. This might sound complicated, but it's just a fancy way of saying we're going to undo the fractional exponent.
By raising both sides to the power of 4/3, we get [(x+9)(3/4)](4/3) = 8^(4/3). On the left side, the exponents (3/4) and (4/3) cancel each other out, leaving us with just (x+9). On the right side, we have 8^(4/3). Now, let's tackle that tricky exponent. Remember, 8^(4/3) means we're taking the cube root of 8 and then raising it to the fourth power. The cube root of 8 is 2, and 2 raised to the fourth power is 16. So, our equation simplifies to x+9 = 16.
We're almost there! Now, we have a simple linear equation. To isolate 'x', we just need to subtract 9 from both sides of the equation. This gives us x = 16 - 9, which simplifies to x = 7. And there you have it! We've successfully solved the equation. The solution is x = 7. But wait, we're not done yet! It's always a good idea to check our answer to make sure it's correct.
Step-by-Step Solution
Let's break down the solution into clear, concise steps:
- Original Equation: 3(x+9)^(3/4) = 24
- Divide both sides by 3: (x+9)^(3/4) = 8
- Raise both sides to the power of 4/3: [(x+9)(3/4)](4/3) = 8^(4/3)
- Simplify: x+9 = 16
- Subtract 9 from both sides: x = 7
Verifying the Solution
To verify our solution, we'll substitute x = 7 back into the original equation: 3(x+9)^(3/4) = 24. Plugging in 7 for x, we get 3(7+9)^(3/4) = 24. Simplifying inside the parentheses, we have 3(16)^(3/4) = 24. Now, we need to evaluate 16^(3/4). This means we're taking the fourth root of 16 and then raising it to the third power. The fourth root of 16 is 2, and 2 raised to the third power is 8. So, we have 3(8) = 24. Multiplying, we get 24 = 24. Voila! Our solution checks out. This confirms that x = 7 is indeed the correct solution to the equation.
Checking our solution is a crucial step in solving any equation, especially those involving fractional exponents or radicals. It helps us catch any potential errors we might have made along the way and ensures that our answer is valid. In this case, our solution worked perfectly, giving us confidence that we've solved the equation correctly.
Potential Pitfalls and How to Avoid Them
When dealing with equations involving fractional exponents, there are a few common pitfalls that students often encounter. Let's discuss these and how to avoid them.
One common mistake is forgetting to raise both sides of the equation to the same power. Remember, the golden rule of algebra is that whatever operation you perform on one side of the equation, you must also perform on the other side to maintain the equality. So, when we raised both sides to the power of 4/3, it was essential to do so on both the left and the right sides. If we had only raised the left side to the power of 4/3, we would have changed the equation and gotten an incorrect solution.
Another pitfall is misinterpreting the meaning of fractional exponents. As we discussed earlier, a fractional exponent like (3/4) represents both a root and a power. It's crucial to understand that the denominator represents the root, and the numerator represents the power. If you mix these up, you'll end up with the wrong answer. To avoid this, always remember that the denominator "dives down" to become the index of the root, and the numerator "floats up" to become the exponent.
Finally, another mistake students make is not checking their solution. As we demonstrated, checking our solution is a critical step in the problem-solving process. It helps us identify any errors we might have made and ensures that our answer is valid. In the case of equations with fractional exponents, checking is especially important because raising to a power can sometimes introduce extraneous solutions (solutions that satisfy the transformed equation but not the original equation). By checking our solution, we can eliminate any extraneous solutions and be confident in our answer.
To avoid these pitfalls, always double-check your work, pay close attention to the meaning of fractional exponents, and remember to check your solution. With practice and careful attention to detail, you'll become a pro at solving these types of equations!
Alternative Approaches and Insights
While we've tackled this equation using a straightforward algebraic approach, it's always beneficial to explore alternative methods and gain deeper insights. Let's consider a slightly different perspective on how we might have approached this problem.
One alternative approach involves rewriting the equation in a slightly different form before we begin solving. Instead of immediately raising both sides to the power of 4/3, we could have first focused on simplifying the term 8^(4/3). As we discussed earlier, this term represents the cube root of 8 raised to the fourth power. We know that the cube root of 8 is 2, so we can rewrite the term as 2^4, which is simply 16. By simplifying this term first, we might have made the subsequent steps a little easier to visualize.
Another insight we can gain from this problem is the connection between fractional exponents and radicals. As we've seen, a fractional exponent is simply a shorthand way of expressing a combination of a root and a power. Understanding this connection can help us solve a wide range of equations involving radicals and exponents. For instance, if we were faced with an equation involving a radical, we could rewrite it using a fractional exponent and then apply the same techniques we've used here to solve it.
Furthermore, this problem highlights the importance of understanding the order of operations. When we evaluated 8^(4/3), we had to remember to take the cube root first and then raise the result to the fourth power. If we had performed these operations in the wrong order, we would have gotten an incorrect answer. Similarly, when we solved for 'x', we had to undo the operations in the reverse order in which they were applied. This principle of reversing the order of operations is fundamental to solving many types of equations.
By exploring these alternative approaches and insights, we not only reinforce our understanding of the problem but also develop a more flexible and adaptable problem-solving mindset. Remember, math isn't just about memorizing formulas and procedures; it's about developing a deep understanding of the underlying concepts and principles.
Real-World Applications of Equations with Fractional Exponents
You might be wondering, "Okay, this is cool and all, but where would I ever use this in real life?" That's a valid question! Equations with fractional exponents, while seemingly abstract, actually have a surprising number of applications in various fields. Let's explore some real-world scenarios where these equations pop up.
One area where fractional exponents are used is in physics, particularly in the study of motion and energy. For example, the period of a simple pendulum (the time it takes for the pendulum to swing back and forth) is related to its length by an equation involving a fractional exponent. Similarly, the velocity of an object falling under the influence of gravity can be described using equations with fractional exponents.
Another field where these equations are relevant is in engineering. When designing structures or machines, engineers often need to calculate the strength and stability of materials. Equations involving fractional exponents are used to model the relationship between stress, strain, and material properties. These calculations are crucial for ensuring that structures can withstand the loads they are subjected to and that machines function safely and efficiently.
Fractional exponents also find applications in finance and economics. For instance, compound interest calculations often involve equations with fractional exponents. These equations are used to determine the future value of an investment or the present value of a loan. Understanding these concepts is essential for making informed financial decisions.
In addition to these specific examples, equations with fractional exponents appear in many other areas of science, technology, engineering, and mathematics (STEM). They are a fundamental tool for modeling and understanding a wide range of phenomena. So, while you might not encounter this exact equation in your daily life, the skills and concepts you've learned in solving it are highly valuable and transferable to many other contexts.
Concluding Thoughts
Alright, guys, we've reached the end of our equation-solving journey! We've successfully tackled the equation 3(x+9)^(3/4) = 24, found the solution x = 7, and even explored some real-world applications of fractional exponents. Hopefully, you now feel more confident and comfortable dealing with these types of equations.
The key takeaway here is that even seemingly complex equations can be solved by breaking them down into smaller, more manageable steps. By systematically undoing the operations affecting the variable, we can isolate it and find its value. Remember to always check your solution to ensure its validity, and don't be afraid to explore alternative approaches and insights.
Math, like any skill, takes practice. So, keep challenging yourself with new problems, and don't get discouraged if you stumble along the way. Every mistake is an opportunity to learn and grow. And who knows, maybe one day you'll be the one explaining these concepts to others! So, keep up the great work, and happy equation-solving!
Final Answer: The final answer is (C) 7