Solving (z-8)^3 = -2 Step-by-Step Solution

In this comprehensive guide, we will delve into the process of solving the equation (z-8)^3 = -2, where z is a real number. Our primary goal is to simplify the solution as much as possible, ensuring that we arrive at the most accurate and concise answer. This exploration is crucial for anyone studying algebra, precalculus, or any field where equation solving is fundamental. We will break down the problem step by step, making it accessible and understandable for students and enthusiasts alike. The process involves algebraic manipulation, understanding cube roots, and simplifying radicals. This equation, while seemingly simple, provides a great opportunity to reinforce key mathematical principles. By the end of this guide, you will not only have the solution to this specific equation but also a better grasp of how to tackle similar problems in the future. Whether you're a student looking to improve your math skills or just someone interested in the beauty of mathematical problem-solving, this guide will provide valuable insights and a clear methodology to follow.

Before we jump into the solution, it's essential to understand the nature of the equation we are dealing with. The equation (z-8)^3 = -2 is a cubic equation, meaning it involves a variable raised to the power of three. The presence of the cube makes it slightly different from quadratic equations (which involve a variable raised to the power of two) or linear equations (where the variable is raised to the power of one). Cubic equations can have up to three solutions, but in this case, we are specifically looking for real solutions – numbers that can be plotted on a number line. The equation is structured such that the cube of the expression (z-8) equals -2. This hints that we'll need to use the concept of cube roots to isolate z. Understanding the properties of cube roots, especially how they handle negative numbers, is crucial in solving this equation. Unlike square roots, cube roots can indeed produce real number outputs from negative inputs. For instance, the cube root of -8 is -2, because (-2) * (-2) * (-2) = -8. This property will be key as we proceed to isolate z and find its value. Recognizing the structure and the implications of the cube root is the first step in effectively solving the equation. Let's now move on to the steps required to find the solution.

To solve the equation (z-8)^3 = -2, we will follow a series of algebraic steps to isolate z. This process involves undoing the operations applied to z, one at a time, until we have z by itself on one side of the equation. Here is a detailed breakdown of the steps:

1. Taking the Cube Root of Both Sides

The first step in solving for z is to eliminate the cube on the left side of the equation. To do this, we take the cube root of both sides. Remember that whatever operation we perform on one side of the equation, we must also perform on the other side to maintain equality. Taking the cube root of (z-8)^3 gives us z-8. On the right side, we have the cube root of -2, which is written as ∛(-2). So, the equation now looks like this:

z - 8 = ∛(-2)

The cube root of a negative number is negative, so ∛(-2) is a real number, unlike the square root of a negative number which would be imaginary. This is an important distinction when solving equations involving radicals.

2. Isolating z

Next, we need to isolate z on the left side of the equation. Currently, z is being subtracted by 8. To undo this subtraction, we add 8 to both sides of the equation. This will effectively cancel out the -8 on the left side and leave z by itself. Adding 8 to both sides, we get:

z = ∛(-2) + 8

Now, z is isolated, and we have an expression for its value. This expression involves the cube root of -2 plus 8. While we could use a calculator to find an approximate decimal value for ∛(-2), it's often preferable to leave the answer in its exact form, especially when dealing with radicals.

3. Simplifying the Cube Root

The cube root of -2, ∛(-2), can be expressed as -∛2. This is because the cube root of -1 is -1, and we can factor out the -1 from under the cube root:

∛(-2) = ∛(-1 * 2) = ∛(-1) * ∛2 = -∛2

This simplification makes the radical part of the solution a bit cleaner and easier to understand.

Now that we have simplified the cube root, we can write the final solution for z. Substituting -∛2 for ∛(-2) in our equation, we get:

z = 8 - ∛2

This is the simplified, exact solution for z. It tells us that z is equal to 8 minus the cube root of 2. This form is the most accurate way to represent the solution, as it avoids any rounding errors that might occur if we used a decimal approximation. The solution z = 8 - ∛2 is a single real number. It represents the point on the number line where the value of (z-8)^3 is exactly -2. To get a sense of where this number lies, we can consider that ∛2 is approximately 1.26, so z is roughly 8 - 1.26, which is about 6.74. However, the exact solution z = 8 - ∛2 is the most precise answer. This solution process demonstrates how to solve cubic equations involving cube roots and how to simplify radicals to present the solution in its most accurate form. By following these steps, you can confidently tackle similar problems in algebra and beyond. The key is to break down the problem into manageable steps, apply the appropriate algebraic operations, and simplify the result whenever possible. This methodical approach not only leads to the correct answer but also deepens your understanding of the underlying mathematical principles.

While the method we've used is the most straightforward for this particular equation, it's worth discussing alternative approaches and considerations that can be applied to similar problems. Understanding these different perspectives can enhance your problem-solving skills and provide you with a broader toolkit for tackling various mathematical challenges.

Numerical Approximation

One alternative method is to use numerical approximation techniques. These methods don't give an exact answer but can provide a decimal approximation to any desired degree of accuracy. For this equation, you could use a calculator or computer software to find the cube root of -2 and then add 8. This approach is particularly useful when an exact solution is difficult or impossible to find, or when a practical application only requires a decimal approximation. Numerical methods include techniques like the Newton-Raphson method, which iteratively refines an initial guess to get closer and closer to the root of the equation. While calculators and software can provide quick approximations, understanding the underlying algorithms helps in appreciating the power and limitations of these tools.

Graphical Approach

Another way to approach the problem is graphically. You can think of the equation (z-8)^3 = -2 as finding the intersection of two graphs: y = (z-8)^3 and y = -2. By plotting these two functions on a graph, the z-coordinate of the intersection point(s) will be the solution(s) to the equation. This method is especially helpful for visualizing the solutions and understanding the behavior of the functions involved. Graphing can also reveal the number of real solutions the equation has. In this case, the cubic function y = (z-8)^3 is a standard cubic curve shifted 8 units to the right. The horizontal line y = -2 intersects this curve at one point, confirming that there is only one real solution. Graphing utilities, both physical and online, make this approach accessible and intuitive.

Complex Solutions

It's also important to note that cubic equations can have up to three solutions, some of which may be complex numbers. In our case, we were specifically asked for real solutions. If we were looking for all solutions, we would need to consider complex roots as well. This involves using the properties of complex numbers and De Moivre's theorem to find the three cube roots of -2. While this is beyond the scope of the current problem, it's an important consideration for a complete understanding of cubic equations. Complex solutions arise from the fact that complex numbers, of the form a + bi where i is the imaginary unit (√-1), extend the number system beyond the real numbers and allow for roots that real numbers alone cannot provide. Exploring complex solutions opens up a rich area of mathematics with applications in various fields, including electrical engineering and quantum mechanics.

General Cubic Equation Solutions

For more complex cubic equations that cannot be easily solved by taking cube roots, there are general methods for finding solutions, such as Cardano's method. This method involves a series of substitutions and algebraic manipulations to reduce the cubic equation to a solvable form. However, Cardano's method can be quite involved and is typically used for more challenging problems. It's a testament to the complexity that cubic equations can exhibit, and it highlights the ingenuity of mathematical techniques developed over centuries to tackle these challenges. The method underscores the power of algebraic manipulation and the quest for general solutions in mathematics. Understanding these broader methods provides a deeper appreciation for the nuances of solving polynomial equations.

In this guide, we have successfully solved the equation (z-8)^3 = -2 for real solutions. We found that the simplified solution is z = 8 - ∛2. This process involved taking the cube root of both sides, isolating z, and simplifying the radical. We also explored alternative methods, such as numerical approximation and graphical approaches, as well as considerations for complex solutions and general cubic equation solving techniques. Solving equations like this is a fundamental skill in mathematics, and the techniques used here can be applied to a wide range of problems. Whether you are a student, an educator, or simply someone who enjoys mathematical challenges, understanding these methods is valuable. Remember, the key to problem-solving in mathematics is to break down the problem into manageable steps, apply the appropriate techniques, and always seek to simplify your answer as much as possible. This methodical approach not only leads to the correct solution but also enhances your understanding of the underlying mathematical principles. Keep practicing, keep exploring, and you'll continue to grow your mathematical abilities. The world of mathematics is vast and fascinating, and every problem solved is a step further on the journey of discovery.

By mastering these techniques and understanding the underlying principles, you can confidently tackle a wide range of mathematical problems and deepen your appreciation for the beauty and power of mathematics. Remember, the journey of mathematical discovery is continuous, and each problem solved is a step forward in your understanding.

Therefore, the final answer is:

z = 8 - ∛2