Solving Equations: A Step-by-Step Guide With Verification

Let's dive into the world of equation-solving! In this comprehensive guide, we will tackle the equation $4 \sqrt[5]{x}+8=24$, providing you with a clear, step-by-step solution. More importantly, we'll emphasize the crucial process of verifying your solutions to ensure accuracy. Solving equations is a fundamental skill in mathematics, and mastering it will open doors to more advanced concepts. So, grab your pen and paper, and let’s get started on this mathematical journey!

Understanding the Equation

Before we jump into solving, let's understand what the equation $4 \sqrt[5]{x}+8=24$ is asking us. The goal here is to find the value(s) of x that make this equation true. This equation involves a fifth root, which might seem intimidating at first, but don't worry! We'll break it down into manageable steps. Recognizing the structure of the equation is the first step towards solving it. We have a variable x under a fifth root, multiplied by a constant, and added to another constant, equaling a final constant. Our mission is to isolate x and unveil its true value. Understanding the components and their relationships is key to choosing the right strategy.

Remember, equations are like puzzles, and each term is a piece. To solve the puzzle, we need to manipulate the pieces while maintaining the balance of the equation. This means whatever operation we perform on one side, we must perform on the other. This principle of balance is the cornerstone of equation solving, ensuring that the equality remains valid throughout the process. As we move forward, keep this principle in mind, and you'll find that even complex equations can be solved with careful attention to detail and a systematic approach. Our first step will be to isolate the term containing the fifth root, paving the way for us to eventually isolate x itself. So, let's begin by addressing the constant term on the left side of the equation.

Step 1: Isolate the Radical Term

The first step in solving this equation is to isolate the term containing the fifth root, which is $4 \sqrt[5]{x}$. To do this, we need to get rid of the $+8$ on the left side of the equation. The opposite of adding 8 is subtracting 8, so we'll subtract 8 from both sides of the equation. This maintains the balance of the equation, ensuring that the equality remains valid. By isolating the radical term, we simplify the equation and bring ourselves closer to isolating x. This step is crucial because it allows us to focus on the part of the equation that directly involves the variable we are trying to solve for.

Subtracting 8 from both sides, we get:

$4 \sqrt[5]{x} + 8 - 8 = 24 - 8$

This simplifies to:

$4 \sqrt[5]{x} = 16$

Now, the radical term is partially isolated, but we still have the coefficient 4 multiplying the fifth root. To completely isolate the radical, we need to eliminate this coefficient. This leads us to our next step, where we'll divide both sides of the equation by 4. This will further simplify the equation and bring us closer to isolating x and finding its value. Remember, each step we take is a deliberate move to unravel the equation and reveal the solution hidden within its structure. So, let's proceed to the next step and continue our journey towards solving for x.

Step 2: Divide by the Coefficient

We now have the equation $4 \sqrt[5]{x} = 16$. To isolate the fifth root, we need to divide both sides of the equation by 4. This eliminates the coefficient multiplying the radical, bringing us closer to isolating x. Remember, whatever operation we perform on one side, we must perform on the other to maintain the balance of the equation. Dividing both sides by the coefficient is a common technique in equation solving, allowing us to simplify the equation and make it easier to manipulate.

Dividing both sides by 4, we get:

$\frac{4 \sqrt[5]{x}}{4} = \frac{16}{4}$

This simplifies to:

$\sqrt[5]{x} = 4$

Now, we have the fifth root of x isolated on one side of the equation. This is a significant step forward, as it allows us to focus on eliminating the radical and solving for x directly. To eliminate the fifth root, we need to perform the inverse operation, which is raising both sides of the equation to the power of 5. This will effectively undo the fifth root and leave us with x isolated. So, let's move on to the next step and raise both sides of the equation to the power of 5 to finally solve for x.

Step 3: Eliminate the Fifth Root

To eliminate the fifth root, we need to raise both sides of the equation $\sqrt[5]{x} = 4$ to the power of 5. This is the inverse operation of taking the fifth root, and it will effectively undo the radical, allowing us to isolate x. Raising both sides to the same power is another way of maintaining the balance of the equation, ensuring that the equality remains valid. This step is crucial because it directly leads us to the solution for x. By eliminating the radical, we transform the equation into a simple form where x is isolated and its value can be easily determined.

Raising both sides to the power of 5, we get:

$(\sqrt[5]{x})^5 = 4^5$

This simplifies to:

$x = 4^5$

Now, we need to calculate $4^5$. This means multiplying 4 by itself five times: $4 * 4 * 4 * 4 * 4$. Performing this calculation will give us the value of x. Once we have the value of x, we'll have a potential solution to the original equation. However, it's crucial to remember that we need to verify this solution to ensure that it is valid. This brings us to the next step, where we'll calculate $4^5$ and then verify our solution by plugging it back into the original equation.

Step 4: Calculate the Value of x

Now, let's calculate $4^5$. As we mentioned earlier, this means multiplying 4 by itself five times: $4 * 4 * 4 * 4 * 4$. Performing this multiplication, we get:

$4^5 = 1024$

So, our potential solution is $x = 1024$. We've arrived at a value for x, but our journey isn't over yet! It's essential to verify our solution by plugging it back into the original equation. This step ensures that our solution is valid and satisfies the equation. Verification is a critical part of the equation-solving process, especially when dealing with radicals, as extraneous solutions can sometimes arise. These are solutions that we obtain through the solving process but do not actually satisfy the original equation.

Therefore, we must always verify our solutions to avoid making mistakes. In the next step, we'll substitute $x = 1024$ back into the original equation and check if both sides of the equation are equal. If they are, then our solution is valid. If not, we'll need to re-examine our steps to identify any potential errors. So, let's proceed to the crucial step of verification and ensure the accuracy of our solution.

Step 5: Verify the Solution

Now, the most crucial step: verifying our solution. We'll substitute $x = 1024$ back into the original equation, $4 \sqrt[5]{x} + 8 = 24$, to see if it holds true. This step is essential to ensure we haven't made any errors and that our solution is valid. Plugging our solution back into the original equation is like putting the final piece in a puzzle – it confirms whether everything fits together correctly.

Substituting $x = 1024$, we get:

$4 \sqrt[5]{1024} + 8 = 24$

Now, we need to simplify the fifth root of 1024. Since we know that $4^5 = 1024$, then $\sqrt[5]{1024} = 4$. Substituting this value, we get:

$4 * 4 + 8 = 24$

Simplifying further:

$16 + 8 = 24$

$24 = 24$

The equation holds true! This confirms that our solution, $x = 1024$, is indeed correct. We've successfully solved the equation and verified our solution, demonstrating a thorough understanding of the equation-solving process. Verification not only confirms the correctness of our answer but also reinforces our understanding of the equation and the steps we took to solve it. Now that we've verified our solution, we can confidently state the answer.

Conclusion

Therefore, the solution to the equation $4 \sqrt[5]{x} + 8 = 24$ is $x = 1024$. We successfully isolated the variable, eliminated the radical, and verified our solution, showcasing a comprehensive understanding of equation-solving techniques. Remember, the key to solving equations is to follow a systematic approach, perform operations carefully, and always verify your solutions. By mastering these skills, you'll be well-equipped to tackle a wide range of mathematical challenges.

Solving equations is a fundamental skill that underpins many areas of mathematics and science. The process we've outlined here – isolating the variable, performing inverse operations, and verifying the solution – is a valuable framework that can be applied to many different types of equations. So, continue practicing and honing your skills, and you'll become a confident and proficient equation solver. For further learning and practice, you might find helpful resources on websites like Khan Academy, which offers a wealth of materials on algebra and equation solving.