Solution To Equation Radical Expression

ADMIN · · 10 min read

Table Of Content

  1. Introduction
    1. Understanding the Equation
    2. The Implications of the Radical
  2. Step-by-Step Solution
    1. Step 1: Simplify the Radical
    2. Step 2: Isolate x
    3. Step 3: Simplify the Right-Hand Side
    4. Step 4: Solve for x
  3. Verification
    1. Substituting the Solution
  4. Conclusion
    1. Key Takeaways
    2. Importance of Equation Solving
    3. Further Exploration

Introduction

In the realm of mathematics, equations serve as fundamental tools for expressing relationships between variables and constants. Solving equations is a core skill, enabling us to determine the unknown values that satisfy the given conditions. This article delves into the process of solving a specific equation involving a radical expression, providing a step-by-step guide to arrive at the solution. We will explore the underlying principles, algebraic manipulations, and potential pitfalls to ensure a comprehensive understanding of the solution process.

The equation at hand is:

(18−x)1=−12\sqrt[1]{\left(\frac{1}{8}-x\right)}=-\frac{1}{2}

This equation presents a unique challenge due to the presence of a radical (specifically, a first root, which is somewhat unusual) and a negative value on the right-hand side. To effectively solve this equation, we will need to carefully consider the properties of radicals and employ appropriate algebraic techniques. This article will break down the solution process into manageable steps, providing clear explanations and justifications for each step taken. By the end of this guide, you will not only understand the solution to this specific equation but also gain valuable insights into solving similar equations involving radicals.

Understanding the Equation

Before diving into the solution, let's first dissect the equation and understand its components. The equation involves a variable, x, which represents the unknown value we aim to find. The left-hand side of the equation features a radical expression, specifically the first root of the expression (1/8 - x). It's crucial to recognize that the first root of any number is simply the number itself. This simplifies the left-hand side, but we must still consider the implications of the negative value on the right-hand side. The right-hand side of the equation is a constant, -1/2. The equation states that the first root of (1/8 - x) is equal to -1/2. Our goal is to isolate x and determine its value.

The Implications of the Radical

While the first root doesn't fundamentally alter the expression inside the radical, it's essential to understand the general principles of radicals. Radicals, especially even roots like square roots or fourth roots, introduce constraints on the values inside them. For instance, the square root of a negative number is not a real number. However, in this case, we are dealing with the first root, which doesn't impose the same restrictions. The first root of any real number is simply the number itself. This simplifies our equation significantly, allowing us to focus on isolating x through algebraic manipulation.

Step-by-Step Solution

Now, let's embark on the journey of solving the equation step by step. We will meticulously analyze each step, providing clear justifications and explanations to ensure a thorough understanding of the solution process.

Step 1: Simplify the Radical

The first step in solving the equation is to simplify the radical expression. As we discussed earlier, the first root of any number is the number itself. Therefore, we can rewrite the equation as:

18−x=−12\frac{1}{8}-x = -\frac{1}{2}

This simplification removes the radical, making the equation easier to manipulate. We now have a linear equation in terms of x. This step is crucial because it transforms the original equation into a more familiar form, which we can solve using standard algebraic techniques. By recognizing that the first root does not change the value of the expression, we eliminate a potential source of confusion and pave the way for further simplification.

Step 2: Isolate x

Our next goal is to isolate x on one side of the equation. To achieve this, we need to eliminate the constant term (1/8) from the left-hand side. We can do this by subtracting 1/8 from both sides of the equation:

18−x−18=−12−18\frac{1}{8}-x - \frac{1}{8}= -\frac{1}{2} - \frac{1}{8}

This operation maintains the equality of the equation while moving us closer to isolating x. The 1/8 on the left-hand side cancels out, leaving us with:

−x=−12−18-x = -\frac{1}{2} - \frac{1}{8}

Now, we need to simplify the right-hand side by finding a common denominator for the fractions.

Step 3: Simplify the Right-Hand Side

To combine the fractions on the right-hand side, we need to find a common denominator. The least common multiple of 2 and 8 is 8. So, we rewrite -1/2 as -4/8:

−x=−48−18-x = -\frac{4}{8} - \frac{1}{8}

Now we can subtract the fractions:

−x=−58-x = -\frac{5}{8}

This step simplifies the equation further, making it easier to solve for x. We now have -x equal to a single fraction.

Step 4: Solve for x

Finally, to solve for x, we need to eliminate the negative sign on the left-hand side. We can do this by multiplying both sides of the equation by -1:

(−1)(−x)=(−1)(−58)(-1)(-x) = (-1)\left(-\frac{5}{8}\right)

This gives us:

x=58x = \frac{5}{8}

Therefore, the solution to the equation is x = 5/8. This is the value of x that satisfies the original equation. We have successfully isolated x and determined its value through a series of algebraic manipulations.

Verification

To ensure the accuracy of our solution, it's always a good practice to verify it by substituting the value of x back into the original equation. This step helps us catch any potential errors in our calculations and provides confidence in our answer.

Substituting the Solution

Let's substitute x = 5/8 back into the original equation:

(18−58)1=−12\sqrt[1]{\left(\frac{1}{8}-\frac{5}{8}\right)}=-\frac{1}{2}

Now, we simplify the expression inside the radical:

(−48)1=−12\sqrt[1]{\left(-\frac{4}{8}\right)}=-\frac{1}{2}

Further simplifying the fraction:

(−12)1=−12\sqrt[1]{\left(-\frac{1}{2}\right)}=-\frac{1}{2}

Since the first root of any number is the number itself, we have:

−12=−12-\frac{1}{2} = -\frac{1}{2}

This confirms that our solution, x = 5/8, is correct. The left-hand side of the equation is equal to the right-hand side when we substitute this value of x. This verification step provides a final check and ensures that our solution is accurate and consistent with the original equation.

Conclusion

In this article, we embarked on a journey to solve the equation:

(18−x)1=−12\sqrt[1]{\left(\frac{1}{8}-x\right)}=-\frac{1}{2}

We meticulously dissected the equation, understanding the implications of the radical and the negative value on the right-hand side. We then employed a step-by-step approach, simplifying the radical, isolating x, and performing algebraic manipulations to arrive at the solution x = 5/8.

Key Takeaways

  • The first root of any number is the number itself.
  • Isolating the variable is a fundamental technique in solving equations.
  • Verifying the solution is crucial to ensure accuracy.

By understanding the principles and techniques outlined in this article, you are well-equipped to tackle similar equations involving radicals and other algebraic expressions. The ability to solve equations is a cornerstone of mathematical proficiency, opening doors to a wide range of applications in science, engineering, and other fields. This comprehensive guide has provided not only the solution to a specific equation but also valuable insights into the broader process of equation solving.

Importance of Equation Solving

Equation solving is not just an academic exercise; it's a fundamental skill with far-reaching applications. From calculating the trajectory of a rocket to modeling the spread of a disease, equations are the language of science and engineering. Mastering the art of solving equations allows us to make predictions, design solutions, and understand the world around us.

The equation we solved in this article, while seemingly simple, exemplifies the core principles of mathematical problem-solving. By breaking down complex problems into manageable steps, applying logical reasoning, and verifying our solutions, we can overcome challenges and gain valuable insights. This process is applicable not only to mathematics but also to other areas of life, fostering critical thinking and problem-solving skills.

Further Exploration

If you found this guide helpful, there are many avenues for further exploration in the realm of equation solving. You can delve into more complex equations involving various types of radicals, polynomials, and trigonometric functions. You can also explore different methods for solving equations, such as graphical methods or numerical techniques.

The world of mathematics is vast and fascinating, and equation solving is just one piece of the puzzle. By continuing to learn and practice, you can deepen your understanding of mathematics and its applications, unlocking new possibilities and expanding your horizons. This journey of mathematical exploration is a rewarding one, filled with intellectual challenges and the satisfaction of discovering new knowledge.

The solution to the equation (18−x)1=−12\sqrt[1]{\left(\frac{1}{8}-x\right)}=-\frac{1}{2} is 58\boxed{\frac{5}{8}}.

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