Solving Equations With Rational Exponents $7x^{\frac{9}{4}}-21=0$

Introduction

In the realm of mathematics, equations with rational exponents often pose a unique challenge. These equations involve variables raised to fractional powers, requiring a systematic approach to solve. This article delves into the intricacies of solving such equations, focusing on the specific example $7x^{\frac{9}{4}}-21=0$. We will explore the step-by-step methodology to isolate the variable, eliminate the rational exponent, and identify all potential solutions. Furthermore, we will emphasize the crucial process of verifying these proposed solutions to ensure their validity, as extraneous solutions can sometimes arise when dealing with rational exponents. Understanding how to solve these types of equations is not only essential for success in algebra and calculus but also provides a foundational skill for various applications in science and engineering. Mastering these techniques allows for a deeper comprehension of mathematical principles and their practical uses.

Understanding Rational Exponents

Before diving into the solution, it's crucial to grasp the concept of rational exponents. A rational exponent is simply an exponent that can be expressed as a fraction, where the numerator indicates the power to which the base is raised, and the denominator represents the root to be taken. For instance, in the term $x^{\frac{9}{4}}$, the exponent $\frac{9}{4}$ signifies that we need to raise $x$ to the power of 9 and then take the fourth root of the result. This understanding is fundamental to manipulating and solving equations involving such exponents. The ability to convert between rational exponents and radical notation is a key skill. For example, $x^{\frac{9}{4}}$ can also be written as $\sqrt[4]{x^9}$. This dual representation allows us to approach problems from different perspectives, sometimes making the solution process more intuitive. Recognizing that rational exponents combine the concepts of powers and roots is essential for simplifying expressions and solving equations effectively. Moreover, understanding the properties of exponents, such as the power of a power rule, product of powers rule, and quotient of powers rule, is crucial for manipulating expressions with rational exponents. These properties provide the tools necessary to simplify complex equations and isolate the variable, ultimately leading to the solution.

Solving the Equation $7x^{\frac{9}{4}}-21=0$

To tackle the equation $7x^{\frac{9}{4}}-21=0$, we will follow a structured approach. Our primary goal is to isolate the term containing the rational exponent, $x^{\frac{9}{4}}$. We begin by adding 21 to both sides of the equation, which gives us $7x^{\frac{9}{4}} = 21$. Next, we divide both sides by 7 to further isolate the term with the rational exponent, resulting in $x^{\frac{9}{4}} = 3$. Now, we need to eliminate the rational exponent. To do this, we raise both sides of the equation to the reciprocal of the exponent, which in this case is $\frac{4}{9}$. This step is crucial because it utilizes the property that $(x^a)^b = x^{a \cdot b}$. Applying this, we get $(x^{\frac{9}{4}})^{\frac{4}{9}} = 3^{\frac{4}{9}}$. Simplifying the left side, we have $x = 3^{\frac{4}{9}}$. This gives us a potential solution for $x$. However, it's essential to remember that when dealing with rational exponents, particularly those with even denominators, we must consider both positive and negative roots. In this specific case, since the denominator of the original exponent is 4 (an even number), we need to be cautious about extraneous solutions. Therefore, we must verify our solution by substituting it back into the original equation.

Step-by-Step Solution

Let's break down the step-by-step solution of the equation $7x^{\frac{9}{4}}-21=0$:

1. Isolate the term with the rational exponent:

   • Add 21 to both sides: $7x^{\frac{9}{4}} = 21$
   • Divide both sides by 7: $x^{\frac{9}{4}} = 3$

2. Eliminate the rational exponent:

   • Raise both sides to the power of $\frac{4}{9}$: $(x^{\frac{9}{4}})^{\frac{4}{9}} = 3^{\frac{4}{9}}$
   • Simplify: $x = 3^{\frac{4}{9}}$

3. Express the solution in radical form (optional):

   • $x = \sqrt[9]{3^4} = \sqrt[9]{81}$

4. Approximate the solution (if needed):

   • $x \approx 1.665$

This detailed breakdown provides a clear path to solving the equation. Each step is logically sequenced, ensuring that the variable is gradually isolated until the solution is obtained. The optional step of expressing the solution in radical form can provide further clarity, especially when dealing with complex exponents. The approximation step is useful for practical applications where a numerical value is required. However, the most critical step is yet to come: verifying the solution.

Checking for Extraneous Solutions

When working with rational exponents, it is crucial to check for extraneous solutions. Extraneous solutions are values that satisfy a transformed equation but not the original equation. This often occurs when raising both sides of an equation to a power, especially when the power involves an even root. To check our solution, $x = 3^{\frac{4}{9}}$, we substitute it back into the original equation: $7x^{\frac{9}{4}}-21=0$. Substituting, we get $7(3^{\frac{4}{9}})^{\frac{9}{4}} - 21 = 0$. Simplifying the exponent, we have $7(3^{\frac{4}{9} \cdot \frac{9}{4}}) - 21 = 0$, which simplifies further to $7(3^1) - 21 = 0$. This gives us $7(3) - 21 = 0$, or $21 - 21 = 0$, which is true. Therefore, $x = 3^{\frac{4}{9}}$ is a valid solution. The importance of this step cannot be overstated. Failing to check for extraneous solutions can lead to incorrect answers and a misunderstanding of the solution set. The process of verification ensures that the solution obtained is not merely a mathematical artifact but a genuine solution to the original problem. This rigorous approach is a hallmark of careful mathematical practice.

Alternative Representations of the Solution

The solution to the equation $7x^{\frac{9}{4}}-21=0$ can be represented in multiple forms, each providing a different perspective on the answer. We found that $x = 3^{\frac{4}{9}}$ is a solution. This form, known as the exponential form, directly expresses $x$ as a power of 3. Another way to represent this solution is in radical form. Recall that $x^{\frac{m}{n}}$ is equivalent to $\sqrt[n]{x^m}$. Applying this to our solution, we can rewrite $3^{\frac{4}{9}}$ as $\sqrt[9]{3^4}$. Simplifying further, we get $\sqrt[9]{81}$. This radical form provides a more visual representation of the root and the power involved. Additionally, we can approximate the solution to a decimal value using a calculator. Evaluating $\sqrt[9]{81}$ or $3^{\frac{4}{9}}$ gives us approximately 1.665. This decimal approximation is useful in practical applications where a numerical answer is required. Understanding these different representations enhances our ability to interpret and apply the solution in various contexts. Each form highlights different aspects of the solution, and being comfortable with these transformations is a valuable skill in mathematics.

Conclusion

In summary, solving equations with rational exponents requires a methodical approach that involves isolating the term with the rational exponent, eliminating the exponent by raising both sides to its reciprocal, and crucially, checking for extraneous solutions. We successfully solved the equation $7x^{\frac{9}{4}}-21=0$, finding the solution to be $x = 3^{\frac{4}{9}}$, which is approximately 1.665. This solution can also be expressed in radical form as $\sqrt[9]{81}$. The process of verifying the solution underscored the importance of this step in ensuring the validity of the answer. Equations with rational exponents are a fundamental concept in algebra and calculus, and mastering their solution techniques is essential for further mathematical studies. The skills developed in solving these equations, such as manipulating exponents and radicals, are transferable to a wide range of mathematical problems. By understanding the underlying principles and practicing diligently, one can confidently tackle equations with rational exponents and appreciate their role in the broader mathematical landscape. This comprehensive guide provides a solid foundation for understanding and solving equations with rational exponents, equipping readers with the knowledge and skills necessary to succeed in this area of mathematics.