Find The Positive Solution Of The Equation.${ 6x^{\frac{9}{2}} + 2 = 3074 }$

Introduction

Solving equations with fractional exponents can be a challenging task, especially when dealing with large numbers. In this article, we will focus on finding the positive solution of the equation $6x^{\frac{9}{2}} + 2 = 3074$. This equation involves a fractional exponent, which requires a different approach to solve. We will use algebraic manipulations and properties of exponents to isolate the variable and find the positive solution.

Understanding the Equation

The given equation is $6x^{\frac{9}{2}} + 2 = 3074$. The first step is to understand the properties of fractional exponents. A fractional exponent is a way of expressing a number as a power of another number. In this case, the exponent is $\frac{9}{2}$, which means that the base $x$ is raised to the power of $\frac{9}{2}$. This can be rewritten as $x^{\frac{9}{2}} = (x^{\frac{1}{2}})^9$.

Isolating the Variable

To isolate the variable $x$, we need to get rid of the constant term $2$ on the left-hand side of the equation. We can do this by subtracting $2$ from both sides of the equation. This gives us $6x^{\frac{9}{2}} = 3072$.

Simplifying the Equation

The next step is to simplify the equation by getting rid of the coefficient $6$. We can do this by dividing both sides of the equation by $6$. This gives us $x^{\frac{9}{2}} = 512$.

Using Properties of Exponents

Now that we have simplified the equation, we can use the properties of exponents to rewrite the equation. We know that $x^{\frac{9}{2}} = (x^{\frac{1}{2}})^9$. Therefore, we can rewrite the equation as $(x^{\frac{1}{2}})^9 = 512$.

Finding the Positive Solution

To find the positive solution of the equation, we need to isolate the variable $x$. We can do this by taking the ninth root of both sides of the equation. This gives us $x^{\frac{1}{2}} = 8$.

Squaring Both Sides

Now that we have isolated the variable $x^{\frac{1}{2}}$, we can square both sides of the equation to get rid of the fractional exponent. This gives us $x = 64$.

Conclusion

In this article, we have found the positive solution of the equation $6x^{\frac{9}{2}} + 2 = 3074$. We used algebraic manipulations and properties of exponents to isolate the variable and find the positive solution. The final answer is $x = 64$.

Step-by-Step Solution

Here is the step-by-step solution to the equation:

1. $6x^{\frac{9}{2}} + 2 = 3074$
2. $6x^{\frac{9}{2}} = 3072$
3. $x^{\frac{9}{2}} = 512$
4. $(x^{\frac{1}{2}})^9 = 512$
5. $x^{\frac{1}{2}} = 8$
6. $x = 64$

Frequently Asked Questions

• What is the positive solution of the equation $6x^{\frac{9}{2}} + 2 = 3074$?
• How do I solve an equation with a fractional exponent?
• What are the properties of exponents that I can use to solve this equation?

Answer

• The positive solution of the equation $6x^{\frac{9}{2}} + 2 = 3074$ is $x = 64$.
• To solve an equation with a fractional exponent, you can use algebraic manipulations and properties of exponents to isolate the variable.
• The properties of exponents that you can use to solve this equation include the power rule, the product rule, and the quotient rule.

Final Answer

The final answer is $x = 64$.

Introduction

Solving equations with fractional exponents can be a challenging task, especially when dealing with large numbers. In this article, we will provide a Q&A section to help you understand the concepts and techniques involved in solving equations with fractional exponents.

Q: What is a fractional exponent?

A: A fractional exponent is a way of expressing a number as a power of another number. It is written in the form $x^{\frac{m}{n}}$, where $m$ and $n$ are integers and $n$ is not equal to zero.

Q: How do I simplify an equation with a fractional exponent?

A: To simplify an equation with a fractional exponent, you can use the properties of exponents to rewrite the equation in a more manageable form. For example, if you have the equation $x^{\frac{9}{2}} = 512$, you can rewrite it as $(x^{\frac{1}{2}})^9 = 512$.

Q: How do I isolate the variable in an equation with a fractional exponent?

A: To isolate the variable in an equation with a fractional exponent, you can use algebraic manipulations and properties of exponents to get rid of the fractional exponent. For example, if you have the equation $6x^{\frac{9}{2}} + 2 = 3074$, you can subtract 2 from both sides of the equation to get $6x^{\frac{9}{2}} = 3072$, and then divide both sides of the equation by 6 to get $x^{\frac{9}{2}} = 512$.

Q: What are some common mistakes to avoid when solving equations with fractional exponents?

A: Some common mistakes to avoid when solving equations with fractional exponents include:

• Not simplifying the equation before solving it
• Not using the properties of exponents to rewrite the equation
• Not isolating the variable correctly
• Not checking the solution to make sure it is valid

Q: How do I check the solution to an equation with a fractional exponent?

A: To check the solution to an equation with a fractional exponent, you can plug the solution back into the original equation and make sure it is true. For example, if you have the equation $x^{\frac{9}{2}} = 512$ and you find the solution $x = 64$, you can plug $x = 64$ back into the original equation to make sure it is true.

Q: What are some real-world applications of solving equations with fractional exponents?

A: Solving equations with fractional exponents has many real-world applications, including:

• Physics: Solving equations with fractional exponents is used to describe the motion of objects with variable acceleration.
• Engineering: Solving equations with fractional exponents is used to design and optimize systems with non-linear behavior.
• Economics: Solving equations with fractional exponents is used to model and analyze economic systems with non-linear behavior.

Q: How can I practice solving equations with fractional exponents?

A: You can practice solving equations with fractional exponents by working through examples and exercises in a textbook or online resource. You can also try solving equations with fractional exponents on your own, using online tools or software to check your work.

Q: What are some common types of equations with fractional exponents?

A: Some common types of equations with fractional exponents include:

• Linear equations with fractional exponents
• Quadratic equations with fractional exponents
• Polynomial equations with fractional exponents
• Rational equations with fractional exponents

Q: How can I use technology to solve equations with fractional exponents?

A: You can use technology, such as graphing calculators or computer algebra systems, to solve equations with fractional exponents. These tools can help you visualize the equation and find the solution.

Q: What are some tips for solving equations with fractional exponents?

A: Some tips for solving equations with fractional exponents include:

• Simplifying the equation before solving it
• Using the properties of exponents to rewrite the equation
• Isolating the variable correctly
• Checking the solution to make sure it is valid

Q: How can I apply the concepts of solving equations with fractional exponents to real-world problems?

A: You can apply the concepts of solving equations with fractional exponents to real-world problems by using the techniques and strategies learned in this article. For example, you can use the properties of exponents to model and analyze systems with non-linear behavior.

Q: What are some common mistakes to avoid when applying the concepts of solving equations with fractional exponents to real-world problems?

A: Some common mistakes to avoid when applying the concepts of solving equations with fractional exponents to real-world problems include:

• Not simplifying the equation before solving it
• Not using the properties of exponents to rewrite the equation
• Not isolating the variable correctly
• Not checking the solution to make sure it is valid

Q: How can I use the concepts of solving equations with fractional exponents to solve problems in physics, engineering, and economics?

A: You can use the concepts of solving equations with fractional exponents to solve problems in physics, engineering, and economics by applying the techniques and strategies learned in this article. For example, you can use the properties of exponents to model and analyze systems with non-linear behavior.

Q: What are some real-world examples of solving equations with fractional exponents?

A: Some real-world examples of solving equations with fractional exponents include:

• Modeling the motion of an object with variable acceleration
• Designing and optimizing systems with non-linear behavior
• Analyzing economic systems with non-linear behavior

Q: How can I use the concepts of solving equations with fractional exponents to solve problems in other fields?

A: You can use the concepts of solving equations with fractional exponents to solve problems in other fields by applying the techniques and strategies learned in this article. For example, you can use the properties of exponents to model and analyze systems with non-linear behavior.

Q: What are some common types of problems that involve solving equations with fractional exponents?

A: Some common types of problems that involve solving equations with fractional exponents include:

• Modeling and analyzing systems with non-linear behavior
• Designing and optimizing systems with non-linear behavior
• Analyzing economic systems with non-linear behavior

Q: How can I use the concepts of solving equations with fractional exponents to solve problems in mathematics?

A: You can use the concepts of solving equations with fractional exponents to solve problems in mathematics by applying the techniques and strategies learned in this article. For example, you can use the properties of exponents to model and analyze systems with non-linear behavior.

Q: What are some real-world examples of solving equations with fractional exponents in mathematics?

A: Some real-world examples of solving equations with fractional exponents in mathematics include:

• Modeling the growth of a population with non-linear behavior
• Analyzing the behavior of a system with non-linear behavior
• Designing and optimizing systems with non-linear behavior

Q: How can I use the concepts of solving equations with fractional exponents to solve problems in other areas of mathematics?

A: You can use the concepts of solving equations with fractional exponents to solve problems in other areas of mathematics by applying the techniques and strategies learned in this article. For example, you can use the properties of exponents to model and analyze systems with non-linear behavior.

Q: What are some common types of problems that involve solving equations with fractional exponents in other areas of mathematics?

A: Some common types of problems that involve solving equations with fractional exponents in other areas of mathematics include:

• Modeling and analyzing systems with non-linear behavior
• Designing and optimizing systems with non-linear behavior
• Analyzing economic systems with non-linear behavior

Q: How can I use the concepts of solving equations with fractional exponents to solve problems in science?

A: You can use the concepts of solving equations with fractional exponents to solve problems in science by applying the techniques and strategies learned in this article. For example, you can use the properties of exponents to model and analyze systems with non-linear behavior.

Q: What are some real-world examples of solving equations with fractional exponents in science?

A: Some real-world examples of solving equations with fractional exponents in science include:

• Modeling the motion of an object with variable acceleration
• Designing and optimizing systems with non-linear behavior
• Analyzing economic systems with non-linear behavior

Q: How can I use the concepts of solving equations with fractional exponents to solve problems in engineering?

A: You can use the concepts of solving equations with fractional exponents to solve problems in engineering by applying the techniques and strategies learned in this article. For example, you can use the properties of exponents to model and analyze systems with non-linear behavior.

Q: What are some real-world examples of solving equations with fractional exponents in engineering?

A: Some real-world examples of solving equations with fractional exponents in engineering include:

• Designing and optimizing systems with non-linear behavior
• Analyzing economic systems with non-linear behavior
• Modeling the behavior of a system with non-linear behavior

Q: How can I use the concepts of solving equations with fractional exponents to solve problems in economics?

A: You can use the concepts of solving equations with fractional exponents to solve problems in economics by applying the techniques and strategies learned in this article. For example, you can use the properties of exponents to model and analyze systems with non-linear behavior.

Q: What are some real-world examples of solving equations with fractional exponents in economics?

A: Some real-world examples of solving equations with fractional exponents in economics include:

• Analyzing economic systems with non-linear behavior
• Modeling the behavior of a system with non-linear behavior
• Design