Solving The Equation Fifth Root Of X+7 Equals -2

In this article, we will delve into solving the equation $\sqrt[5]{x+7}=-2$. This equation involves a fifth root, which might seem intimidating at first, but we will break it down step by step to arrive at the correct solution. Understanding how to solve such equations is crucial for mastering algebra and related mathematical concepts. Many real-world applications, ranging from physics to engineering, require the ability to manipulate and solve equations involving radicals. Our focus will be on providing a clear, detailed explanation to ensure that the process is easily understood and can be applied to similar problems. The main goal is to clarify the steps needed to isolate the variable x and determine whether the equation has a valid solution.

Before we dive into the solution, let's first understand the equation $\sqrt[5]{x+7}=-2$. This equation states that the fifth root of the quantity x + 7 is equal to -2. A fifth root is a number that, when raised to the power of 5, yields the given quantity. In other words, we are looking for a number that, when multiplied by itself five times, results in x + 7. The presence of a negative value on the right-hand side of the equation adds an interesting twist, which we'll address in detail. It's important to note that odd roots (like fifth roots) can yield negative results, unlike even roots (like square roots), which typically yield non-negative results in the realm of real numbers. Understanding the properties of radicals and roots is fundamental to solving this type of equation correctly. This understanding allows us to anticipate potential issues, such as extraneous solutions, which might arise during the solution process. Grasping the nature of roots and their behavior with negative numbers is essential for accuracy and confidence in algebraic manipulations.

To solve the equation $\sqrt[5]{x+7}=-2$, we need to isolate x. The primary operation we'll use is raising both sides of the equation to the power of 5. This will eliminate the fifth root and allow us to work directly with x + 7. Here's the step-by-step process:

1. Raise both sides to the power of 5:

   $(\sqrt[5]{x+7})^5 = (-2)^5$

   This step is crucial because it reverses the operation of taking the fifth root. When we raise a fifth root to the power of 5, we are left with the original expression inside the root. On the right side, we need to calculate (-2) raised to the power of 5. Understanding the properties of exponents is essential here, especially when dealing with negative bases and odd exponents.

2. Simplify both sides:

   $x+7 = -32$

   Here, $(\sqrt[5]{x+7})^5$ simplifies to x + 7, as raising a fifth root to the power of 5 cancels out the root. On the right side, (-2)^5 is calculated as -2 * -2 * -2 * -2 * -2, which equals -32. The negative sign is retained because we are raising a negative number to an odd power. This simplification brings us closer to isolating x and finding its value. It is a straightforward but important step in the solution process, demonstrating the power of inverse operations in algebra.

3. Isolate x:

   To isolate x, we subtract 7 from both sides of the equation:

   $x+7-7 = -32-7$

   $x = -39$

   This step involves a simple subtraction to isolate x on one side of the equation. Subtracting 7 from both sides maintains the equation's balance and brings us to the solution x = -39. This isolation of x is the goal of solving any algebraic equation, and it's achieved here through basic arithmetic. The result, x = -39, is our candidate solution, but it's crucial to verify it to ensure it satisfies the original equation.

4. Verify the solution:

   To ensure our solution is correct, we substitute x = -39 back into the original equation:

   $\sqrt[5]{-39+7} = -2$

   $\sqrt[5]{-32} = -2$

   The verification step is a critical part of solving any equation, particularly those involving radicals. Substituting x = -39 into the original equation, we get $\sqrt[5]{-39+7} = \sqrt[5]{-32}$. The fifth root of -32 is indeed -2, since (-2)^5 = -32. This confirms that our solution is valid and not an extraneous solution. Extraneous solutions can sometimes arise when dealing with radical equations, so this verification step ensures accuracy and completeness in our solution process. The fact that -2 equals -2 confirms that our value for x is the correct solution.

After carefully solving the equation $\sqrt[5]{x+7}=-2$, we have determined that the solution is x = -39. We achieved this by raising both sides of the equation to the power of 5, simplifying, and isolating x. It's crucial to always verify the solution, particularly in equations involving radicals, to ensure that the solution is not extraneous. This problem highlights the importance of understanding the properties of radicals and exponents in solving algebraic equations. By mastering these techniques, you can confidently tackle a wide range of algebraic challenges. Remember, the key to solving mathematical problems is a systematic approach, attention to detail, and a solid understanding of the fundamental principles involved. Consistent practice and a thorough verification process will ensure accuracy and build confidence in your problem-solving abilities. The final solution x = -39 not only answers the question but also reinforces the method for solving similar radical equations.