Solution To 3(x+9)^(3/4) = 24 - A Step-by-Step Guide

This article provides a step-by-step solution to the equation $3(x+9)^{\frac{3}{4}}=24$, a common type of problem encountered in mathematics, particularly in algebra. Understanding how to solve such equations is crucial for students and anyone dealing with mathematical problems in various fields. This article aims to provide a clear, concise, and comprehensive guide to solving this specific equation, as well as offering insights into the broader concepts involved. Solving exponential equations often involves manipulating exponents and radicals, and this example provides an excellent opportunity to practice these skills.

Understanding the Problem

Before diving into the solution, it's important to understand the problem. The equation $3(x+9)^{\frac{3}{4}}=24$ involves a variable, $x$, raised to a fractional exponent. Fractional exponents combine the concepts of powers and roots. In this case, the exponent $\frac{3}{4}$ means we are taking the fourth root of $(x+9)$ and then cubing the result. The equation is set equal to 24, and our goal is to find the value(s) of $x$ that satisfy this equation. The question requires us to isolate the variable x by performing a series of algebraic operations. To successfully solve this, we need to understand the order of operations and how to reverse them to isolate x. Grasping the fundamentals of exponents and radicals is crucial for solving this type of problem. Furthermore, understanding the properties of equality, such as the addition, subtraction, multiplication, and division properties, is essential for manipulating the equation while maintaining its balance. This particular equation is a good example of how these concepts come together in a practical problem-solving scenario. We will break down each step of the solution process to ensure clarity and understanding. By the end of this article, you should not only be able to solve this specific equation but also have a better understanding of how to approach similar problems involving fractional exponents and radicals. Remember, practice is key to mastering these skills, so be sure to work through similar examples to solidify your understanding. The concepts involved here are fundamental to many areas of mathematics and science, making it a valuable skill to develop.

Step-by-Step Solution

To solve the equation $3(x+9)^{\frac{3}{4}}=24$, we'll follow these steps:

1. Isolate the Exponential Term

The first step is to isolate the term with the fractional exponent, which is $(x+9)^{\frac{3}{4}}$. To do this, we divide both sides of the equation by 3:

$$(x+9)^{\frac{3}{4}} = \frac{24}{3}$$

$$(x+9)^{\frac{3}{4}} = 8$$

This step simplifies the equation and brings us closer to isolating the variable x. The principle behind this step is the division property of equality, which states that dividing both sides of an equation by the same non-zero number maintains the equality. Isolating the exponential term is a common strategy when dealing with equations involving exponents or radicals. It allows us to focus on the core part of the equation that involves the variable, making subsequent steps easier to perform. By isolating the exponential term, we are essentially undoing the multiplication by 3, which was the last operation applied to the term $(x+9)^{\frac{3}{4}}$. This step sets the stage for the next operation, which will involve dealing with the fractional exponent. Understanding the importance of isolating the variable or the term containing the variable is crucial in solving algebraic equations. It's a foundational skill that will be used repeatedly in more complex problems. Always look for opportunities to simplify the equation by isolating terms, as it often makes the remaining steps clearer and more manageable. This initial step is a critical building block in the overall solution process.

2. Eliminate the Fractional Exponent

The next step is to eliminate the fractional exponent $\frac{3}{4}$. To do this, we raise both sides of the equation to the reciprocal of the exponent, which is $\frac{4}{3}$:

$$[(x+9)^{\frac{3}{4}}]^{\frac{4}{3}} = 8^{\frac{4}{3}}$$

When raising a power to another power, we multiply the exponents. So, on the left side, $\frac{3}{4} \cdot \frac{4}{3} = 1$, which leaves us with:

$$x+9 = 8^{\frac{4}{3}}$$

On the right side, we need to evaluate $8^{\frac{4}{3}}$. This means taking the cube root of 8 and then raising the result to the fourth power. The cube root of 8 is 2, so we have:

$$8^{\frac{4}{3}} = (8^{\frac{1}{3}})^4 = 2^4 = 16$$

Thus, the equation becomes:

$$x+9 = 16$$

This step is crucial because it removes the fractional exponent, allowing us to solve for x directly. The key concept here is the power of a power rule, which states that $(a^m)^n = a^{m \cdot n}$. By raising both sides to the reciprocal of the exponent, we effectively cancel out the fractional exponent on the left side. Understanding how to work with fractional exponents is essential for solving equations like this. Remember that a fractional exponent represents both a root and a power. In this case, $\frac{4}{3}$ means taking the cube root (denominator) and then raising to the fourth power (numerator). The order in which you perform these operations doesn't matter, but it's often easier to take the root first to keep the numbers smaller. This step demonstrates the power of using inverse operations to isolate the variable. Just as we used division to undo multiplication in the previous step, here we are using exponentiation to undo the fractional exponent. This is a common strategy in algebra: identify the operations that have been applied to the variable and then use their inverses to undo them. This logical approach is fundamental to solving a wide range of algebraic equations.

3. Isolate x

Finally, to isolate x, we subtract 9 from both sides of the equation:

$$x+9-9 = 16-9$$

$$x = 7$$

Therefore, the solution to the equation is $x = 7$. This is a straightforward step that completes the process of isolating the variable x. Subtracting 9 from both sides is justified by the subtraction property of equality, which ensures that the equation remains balanced. This final step underscores the importance of systematically undoing operations to isolate the variable. By performing the inverse operations in the correct order, we have successfully solved for x. Isolating the variable is a fundamental goal in solving any algebraic equation, and this example clearly demonstrates how to achieve this through a series of logical steps. The solution $x = 7$ is the value that, when substituted back into the original equation, will make the equation true. This can be verified by plugging 7 back into the original equation: $3(7+9)^{\frac{3}{4}} = 3(16)^{\frac{3}{4}} = 3(2^4)^{\frac{3}{4}} = 3(2^3) = 3(8) = 24$. This verification step is a good practice to ensure that the solution is correct. It reinforces the understanding of the equation and the solution process.

Answer

The solution to the equation $3(x+9)^{\frac{3}{4}}=24$ is:

C. 7

This final answer confirms the result obtained through our step-by-step solution. The process of solving this equation highlights several important algebraic techniques, including isolating terms, working with fractional exponents, and applying the properties of equality. The correct answer, 7, satisfies the original equation, demonstrating the validity of the solution process. Understanding the steps involved in solving this type of equation is crucial for building a strong foundation in algebra. It's not just about arriving at the correct answer but also about understanding the underlying principles and techniques. By breaking down the problem into smaller, manageable steps, we can effectively solve complex equations. The ability to solve equations like this is essential for success in higher-level mathematics courses and in many real-world applications. This example provides a valuable opportunity to practice and reinforce these skills. The solution process emphasizes the importance of careful and methodical work. Each step must be performed accurately to arrive at the correct answer. This attention to detail is a key attribute of successful problem-solvers in mathematics. The final answer, 7, is not just a number; it represents the value of x that makes the equation true. This understanding of the meaning of a solution is crucial for applying mathematics to real-world problems.

Conclusion

Solving the equation $3(x+9)^{\frac{3}{4}}=24$ demonstrates the application of key algebraic principles, including isolating terms, manipulating exponents, and applying the properties of equality. The step-by-step solution provides a clear and concise method for solving this type of problem. The process involved in solving this equation is a microcosm of the broader mathematical problem-solving process. It involves understanding the problem, developing a strategy, executing the strategy, and verifying the solution. These are valuable skills that can be applied to a wide range of problems, both in mathematics and in other fields. Mastering these techniques is essential for anyone pursuing further studies in mathematics or science. The ability to confidently solve equations is a fundamental skill that opens doors to more advanced concepts and applications. This example serves as a reminder that even seemingly complex problems can be solved by breaking them down into smaller, more manageable steps. The key is to understand the underlying principles and apply them systematically. The solution to this equation, 7, is a testament to the power of algebraic techniques. It's a reminder that mathematics is a logical and consistent system, and with the right tools and knowledge, we can solve a wide range of problems. The journey of solving this equation is as important as the destination. The skills and understanding gained along the way are what truly matter. So, continue to practice and explore the world of mathematics, and you'll find that even the most challenging problems can be conquered with the right approach.