Solving X^(-3) = 8 A Step By Step Guide

In this article, we will delve into solving the equation x^(-3) = 8. This equation involves a negative exponent, which might seem tricky at first, but with a clear understanding of exponent rules, we can find the value of x step by step. Our goal is to transform the equation into a more manageable form, isolate x, and arrive at the correct solution. This type of problem is common in algebra and is a fundamental concept for more advanced mathematical topics. We will discuss the properties of exponents, particularly negative exponents and fractional exponents, to provide a comprehensive understanding of the solution process. The correct answer is crucial, but understanding the how and why behind the solution is even more important. This article is designed to not only give the answer but also to equip you with the knowledge to tackle similar problems confidently.

Before diving into the solution, let's discuss negative exponents. A negative exponent indicates a reciprocal. Specifically, x^(-n) is the same as 1/x^n. This property is crucial for solving equations like x^(-3) = 8. When we see a negative exponent, it means we are dealing with the reciprocal of the base raised to the positive exponent. Understanding this concept is the first step in simplifying and solving the given equation. Many students initially struggle with negative exponents, but once the reciprocal relationship is understood, the problems become much more approachable. This understanding not only helps in solving equations but also in simplifying expressions and working with scientific notation. The key takeaway here is that a negative exponent does not result in a negative number; it results in a reciprocal. We will use this principle extensively in the following sections to transform the given equation into a solvable form.

Now that we understand negative exponents, let's apply this knowledge to the equation x^(-3) = 8. According to the rule of negative exponents, we can rewrite x^(-3) as 1/x^3. So, the equation becomes 1/x^3 = 8. This transformation is a significant step because it changes the equation from a somewhat abstract form to a more concrete form. The next step involves isolating x^3. To do this, we can take the reciprocal of both sides of the equation. This gives us x^3 = 1/8. We now have a much simpler equation to work with. Recognizing and applying this initial transformation is often the key to solving equations with negative exponents. This step allows us to move from dealing with a reciprocal variable to dealing with a variable raised to a positive power, which is a more familiar form for most students. In the next section, we will explore how to solve for x when it is raised to a power.

We've transformed the equation to x^3 = 1/8. To solve for x, we need to find the number that, when multiplied by itself three times, equals 1/8. This is where the concept of cube roots comes in. We need to take the cube root of both sides of the equation. The cube root of x^3 is simply x. The cube root of 1/8 can be found by considering the cube roots of the numerator and denominator separately. The cube root of 1 is 1, and the cube root of 8 is 2. Therefore, the cube root of 1/8 is 1/2. This gives us the solution x = 1/2. Understanding cube roots is essential for solving equations where the variable is raised to the power of 3. It's also important to remember that every positive number has a real cube root. In this case, we found that 1/2 multiplied by itself three times equals 1/8, confirming our solution. This process of taking roots to solve for a variable is a common technique in algebra and is widely applicable to various types of equations.

It's always a good practice to verify our solution. To verify that x = 1/2 is the correct solution, we substitute it back into the original equation, x^(-3) = 8. Substituting x = 1/2, we get (1/2)^(-3). Using the rule of negative exponents, this is equal to (2/1)^3, which simplifies to 2^3. 2^3 is 2 * 2 * 2, which equals 8. Since 8 = 8, our solution x = 1/2 is correct. Verification is a crucial step in problem-solving because it helps us catch any potential errors. By substituting the solution back into the original equation, we ensure that it satisfies the equation. This process builds confidence in our answer and reinforces our understanding of the concepts involved. In more complex problems, verification can be more involved, but the principle remains the same: to ensure the solution is consistent with the initial conditions of the problem.

In this article, we successfully solved the equation x^(-3) = 8 and found that x = 1/2. We began by understanding the property of negative exponents, which allowed us to transform the equation into a more manageable form. We then took the cube root of both sides to isolate x and find the solution. Finally, we verified our solution by substituting it back into the original equation. This problem highlights the importance of understanding exponent rules and the relationship between exponents and roots. By mastering these concepts, you can tackle a wide range of algebraic problems with confidence. Remember, the key to success in mathematics is not just finding the correct answer but also understanding the underlying principles and processes. This comprehensive approach not only improves problem-solving skills but also builds a solid foundation for more advanced mathematical studies. Keep practicing and applying these concepts, and you'll find that even complex problems become more approachable.

Final Answer: The final answer is (C).