Solving For X: $\sqrt[3]{x+28}=4$ - A Step-by-Step Guide

Welcome to this comprehensive guide on solving the equation $\sqrt[3]{x+28}=4$. In this article, we will break down the steps required to find the value of x. Whether you're a student tackling algebra or just someone looking to sharpen your math skills, this guide will provide you with a clear and easy-to-follow solution. So, let's dive in and unravel this equation together!

Understanding the Equation

Before we jump into the solution, let's make sure we understand the equation we're dealing with: $\sqrt[3]{x+28}=4$. This equation involves a cube root, which might seem intimidating at first, but don't worry! We'll take it one step at a time. At its core, this equation is asking: "What number, when 28 is added to it, has a cube root of 4?" To answer this, we need to isolate x, and to do that, we'll need to reverse the operations applied to it.

When solving algebraic equations, it's crucial to grasp the underlying concepts. In this specific equation, $\sqrt[3]{x+28}=4$, we encounter a cube root. Understanding cube roots is fundamental to solving this problem efficiently. A cube root is a number that, when multiplied by itself three times, yields the original number. For example, the cube root of 8 is 2 because $2 \times 2 \times 2 = 8$. Similarly, the cube root of 27 is 3, and so on. Recognizing this relationship between a number and its cube root allows us to reverse the operation and isolate the variable we're trying to solve for. In our equation, we're dealing with the cube root of the expression $x + 28$. Our goal is to undo this cube root operation, and to do that, we'll use the inverse operation, which is cubing both sides of the equation. This is a standard technique in algebra and is essential for solving equations involving radicals.

Now, let's talk about why isolating x is so important. In any algebraic equation, the variable x represents an unknown value that we're trying to find. The equation provides a relationship between x and other numbers or expressions. Our task is to manipulate the equation in such a way that we get x by itself on one side, revealing its value. This process often involves performing inverse operations, such as adding or subtracting numbers, multiplying or dividing, and in this case, dealing with cube roots. The key idea is to maintain balance in the equation. Whatever operation we perform on one side, we must also perform on the other side to ensure the equality remains true. By systematically isolating x, we can unravel the equation and determine the value that satisfies the given relationship. This skill is crucial in algebra and forms the basis for solving more complex mathematical problems.

Step-by-Step Solution

Step 1: Eliminate the Cube Root

The first step to solving for x is to eliminate the cube root. To do this, we'll cube both sides of the equation. Cubing a cube root cancels out the radical, leaving us with just the expression inside.

So, we have:

$(\sqrt[3]{x+28})^3 = 4^3$

This simplifies to:

$x + 28 = 64$

Step 2: Isolate x

Now that we've eliminated the cube root, we need to isolate x. This means getting x by itself on one side of the equation. To do this, we'll subtract 28 from both sides:

$x + 28 - 28 = 64 - 28$

This simplifies to:

$x = 36$

And there we have it! The value of x that satisfies the equation is 36.

Detailed Explanation of Each Step

Step 1: Eliminate the Cube Root

This step is crucial because it removes the radical, which is the main obstacle in solving the equation. When we cube both sides of the equation, we are essentially applying the inverse operation of the cube root. The cube root function and the cubing function are inverses of each other, meaning they cancel each other out. This is similar to how addition and subtraction are inverse operations, or how multiplication and division are inverse operations. By cubing both sides, we transform the equation from one that involves a cube root to a simpler linear equation. This transformation makes the equation much easier to solve because we're left with a straightforward algebraic expression. It's important to remember that whatever operation we perform on one side of the equation, we must perform on the other side to maintain the balance and ensure the equality remains valid. In this case, cubing both sides allows us to eliminate the cube root on the left side while cubing the number on the right side gives us a new numerical value to work with.

Step 2: Isolate x

Once we've eliminated the cube root, the next step is to isolate x. This means getting x by itself on one side of the equation. To do this, we need to undo any operations that are being applied to x. In this case, x is being added to 28. To undo this addition, we perform the inverse operation, which is subtraction. We subtract 28 from both sides of the equation. Subtracting 28 from the left side cancels out the +28, leaving x by itself. Subtracting 28 from the right side gives us a new numerical value. By isolating x, we are essentially solving for its value. The goal is to rewrite the equation in the form x = a number. This number represents the solution to the equation, the value that makes the equation true. In this step, we use basic algebraic principles to manipulate the equation and arrive at the solution. The process of isolating x is a fundamental skill in algebra and is used in solving a wide range of equations.

Verifying the Solution

It's always a good practice to verify your solution to make sure it's correct. To do this, we'll substitute x = 36 back into the original equation:

$\sqrt[3]{36+28} = 4$

$\sqrt[3]{64} = 4$

$4 = 4$

Since the equation holds true, our solution is correct!

The Importance of Verifying Solutions

Verifying solutions is a crucial step in the problem-solving process in mathematics. It serves as a check to ensure that the value we've found for the variable actually satisfies the original equation. This is particularly important in algebra, where we often manipulate equations to isolate variables and solve for unknowns. The process of solving equations involves a series of steps, and it's possible to make a mistake along the way, such as an arithmetic error or an incorrect application of an algebraic rule. By verifying the solution, we can catch these mistakes and avoid arriving at a wrong answer. The verification process involves substituting the value we've found for the variable back into the original equation. If the equation holds true, meaning that both sides of the equation are equal, then we can be confident that our solution is correct. If the equation does not hold true, it indicates that there is an error in our solution, and we need to go back and recheck our steps to find the mistake. Verifying solutions not only helps us to ensure accuracy but also reinforces our understanding of the equation and the solution process. It's a valuable habit to develop in mathematics, as it promotes carefulness and attention to detail.

In the context of our equation, $\sqrt[3]{x+28}=4$, substituting x = 36 back into the equation is a straightforward process. We replace x with 36 and then simplify the expression on the left side. This involves adding 36 and 28, which gives us 64, and then taking the cube root of 64. If the cube root of 64 is indeed 4, then our solution is verified. This verification step provides us with assurance that we have solved the equation correctly. It's a simple yet effective way to confirm our answer and avoid errors.

Alternative Methods

While cubing both sides is the most straightforward method for solving this equation, there are other approaches you could take. For instance, you could think about what number, when cubed, equals 64 (since we know $x + 28$ must equal 64). However, for equations like this, the algebraic method is generally the most efficient and reliable.

Exploring Different Problem-Solving Approaches

In mathematics, there is often more than one way to solve a problem. Exploring different problem-solving approaches can be beneficial for several reasons. First, it can deepen our understanding of the underlying mathematical concepts. When we approach a problem from different angles, we gain a more comprehensive view of the relationships between the variables and the operations involved. This can lead to a more intuitive grasp of the material. Second, exploring alternative methods can improve our problem-solving skills. By considering different strategies, we develop the ability to adapt our approach based on the specific characteristics of the problem. This flexibility is a valuable asset in mathematics and beyond. Third, sometimes one method may be more efficient or easier to apply than another, depending on the problem. By knowing multiple methods, we can choose the one that is most suitable for the given situation.

In the context of the equation $\sqrt[3]{x+28}=4$, while cubing both sides is the most direct method, we could also consider the properties of cube roots and perfect cubes. We know that the cube root of a number is the value that, when multiplied by itself three times, equals the original number. So, we could think about what number cubed equals 64, since $x + 28$ must equal 64. This approach involves a more conceptual understanding of cube roots and can be helpful for building intuition. However, for more complex equations, the algebraic method of cubing both sides is generally more reliable and efficient. It's important to have a variety of problem-solving tools at our disposal so that we can tackle different types of equations with confidence.

Conclusion

In this article, we've successfully solved the equation $\sqrt[3]{x+28}=4$, finding that x = 36. We walked through each step, from eliminating the cube root to isolating x, and we even verified our solution. Remember, practice makes perfect, so keep working on these types of problems to build your algebra skills!

If you're interested in learning more about algebraic equations and problem-solving techniques, I highly recommend checking out resources like Khan Academy's Algebra Section. They offer a wealth of tutorials and exercises to help you master these concepts.