Solve For The Value Of $x$:$(1) \cdot \left(\frac{1}{8}\right)^{4x-9} = 64^{3x+1}$

Introduction

Exponential equations are a fundamental concept in mathematics, and solving them requires a deep understanding of the underlying principles. In this article, we will focus on solving a specific type of exponential equation, which involves a variable in the exponent. We will use the given equation as a case study to illustrate the steps involved in solving such equations.

The Given Equation

The given equation is:

$$(1) \cdot \left(\frac{1}{8}\right)^{4x-9} = 64^{3x+1}$$

Our goal is to solve for the value of $x$.

Step 1: Simplify the Right-Hand Side

To simplify the right-hand side, we can rewrite $64$ as $8^3$. This gives us:

$$64^{3x+1} = (8^3)^{3x+1} = 8^{9x+3}$$

Now, the equation becomes:

$$(1) \cdot \left(\frac{1}{8}\right)^{4x-9} = 8^{9x+3}$$

Step 2: Express Both Sides with the Same Base

To make it easier to compare the two sides, we can express both sides with the same base. We can rewrite $\frac{1}{8}$ as $8^{-1}$. This gives us:

$$(1) \cdot (8^{-1})^{4x-9} = 8^{9x+3}$$

Now, we can simplify the left-hand side using the property of exponents that states $(a^m)^n = a^{mn}$:

$$(1) \cdot 8^{-(4x-9)} = 8^{9x+3}$$

Step 3: Equate the Exponents

Since the bases are the same, we can equate the exponents:

$$-(4x-9) = 9x+3$$

Step 4: Solve for $x$

To solve for $x$, we can start by simplifying the equation:

$$-4x+9 = 9x+3$$

Next, we can add $4x$ to both sides to get:

$$9 = 13x+3$$

Then, we can subtract $3$ from both sides to get:

$$6 = 13x$$

Finally, we can divide both sides by $13$ to get:

$$x = \frac{6}{13}$$

Conclusion

In this article, we have solved a specific type of exponential equation that involves a variable in the exponent. We have used the given equation as a case study to illustrate the steps involved in solving such equations. By following these steps, we have been able to solve for the value of $x$.

Tips and Tricks

• When solving exponential equations, it's essential to simplify the equation by expressing both sides with the same base.
• Use the property of exponents that states $(a^m)^n = a^{mn}$ to simplify the equation.
• Equate the exponents since the bases are the same.
• Solve for the variable by simplifying the equation and isolating the variable.

Common Mistakes

• Failing to simplify the equation by expressing both sides with the same base.
• Not using the property of exponents to simplify the equation.
• Not equating the exponents since the bases are the same.
• Not solving for the variable by simplifying the equation and isolating the variable.

Real-World Applications

Exponential equations have numerous real-world applications, including:

• Modeling population growth and decline
• Calculating compound interest
• Analyzing the spread of diseases
• Predicting the behavior of complex systems

By understanding how to solve exponential equations, we can better analyze and model real-world phenomena.

Final Thoughts

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Introduction

In our previous article, we discussed how to solve exponential equations that involve a variable in the exponent. We used a specific equation as a case study to illustrate the steps involved in solving such equations. In this article, we will provide a Q&A guide to help you better understand the concepts and techniques involved in solving exponential equations.

Q: What is an exponential equation?

A: An exponential equation is an equation that involves a variable in the exponent. It is a type of equation that can be written in the form $a^x = b$, where $a$ and $b$ are constants and $x$ is the variable.

Q: How do I simplify an exponential equation?

A: To simplify an exponential equation, you can express both sides with the same base. This can be done by rewriting the equation in a form that has the same base on both sides. For example, if the equation is $2^x = 8^y$, you can rewrite it as $2^x = (2^3)^y = 2^{3y}$.

Q: What is the property of exponents that I should use to simplify an exponential equation?

A: The property of exponents that you should use to simplify an exponential equation is $(a^m)^n = a^{mn}$. This property states that when you raise a power to a power, you multiply the exponents.

Q: How do I equate the exponents in an exponential equation?

A: To equate the exponents in an exponential equation, you can set the exponents equal to each other. For example, if the equation is $2^x = 2^{3y}$, you can equate the exponents by setting $x = 3y$.

Q: What are some common mistakes to avoid when solving exponential equations?

A: Some common mistakes to avoid when solving exponential equations include:

• Failing to simplify the equation by expressing both sides with the same base.
• Not using the property of exponents to simplify the equation.
• Not equating the exponents since the bases are the same.
• Not solving for the variable by simplifying the equation and isolating the variable.

Q: What are some real-world applications of exponential equations?

A: Exponential equations have numerous real-world applications, including:

• Modeling population growth and decline
• Calculating compound interest
• Analyzing the spread of diseases
• Predicting the behavior of complex systems

Q: How can I practice solving exponential equations?

A: You can practice solving exponential equations by working through examples and exercises. You can also try solving real-world problems that involve exponential equations.

Q: What are some tips for solving exponential equations?

A: Some tips for solving exponential equations include:

• Simplifying the equation by expressing both sides with the same base.
• Using the property of exponents to simplify the equation.
• Equating the exponents since the bases are the same.
• Solving for the variable by simplifying the equation and isolating the variable.

Conclusion

In this article, we have provided a Q&A guide to help you better understand the concepts and techniques involved in solving exponential equations. We have covered topics such as simplifying exponential equations, equating exponents, and avoiding common mistakes. We hope that this guide has been helpful in your studies and that you will be able to apply the concepts and techniques to real-world problems.

Additional Resources

• Khan Academy: Exponential Equations
• Mathway: Exponential Equations
• Wolfram Alpha: Exponential Equations

Final Thoughts

Solving exponential equations requires a deep understanding of the underlying principles. By following the steps outlined in this article and practicing with examples and exercises, you can become proficient in solving exponential equations and apply the concepts and techniques to real-world problems.