Solve For { X$} : : : { 9^6 = \left(3^2\right)^{2x-2} \} Possible Solutions:A. { X = \frac{5}{2} $}$B. { X = 2 $}$C. { X = \log 9 $}$D. { X = 4 $}$

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Introduction

Exponential equations are a fundamental concept in mathematics, and solving them requires a deep understanding of algebraic manipulations and properties of exponents. In this article, we will focus on solving a specific exponential equation, 96=(32)2x−29^6 = \left(3^2\right)^{2x-2}, and explore the possible solutions.

Understanding Exponents

Before diving into the solution, it's essential to understand the properties of exponents. The exponent of a number is the power to which the number is raised. For example, in the expression aba^b, aa is the base and bb is the exponent. When we multiply two numbers with the same base, we add their exponents. For instance, am⋅an=am+na^m \cdot a^n = a^{m+n}.

Solving the Exponential Equation

Now, let's focus on solving the given exponential equation, 96=(32)2x−29^6 = \left(3^2\right)^{2x-2}. To simplify the equation, we can rewrite 99 as 323^2, since 9=329 = 3^2. The equation becomes:

(32)6=(32)2x−2\left(3^2\right)^6 = \left(3^2\right)^{2x-2}

Using the property of exponents that states (am)n=amn(a^m)^n = a^{mn}, we can simplify the equation further:

32⋅6=32(2x−2)3^{2 \cdot 6} = 3^{2(2x-2)}

This simplifies to:

312=34x−43^{12} = 3^{4x-4}

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

12=4x−412 = 4x-4

Solving for x

Now, we need to solve for xx. To do this, we can add 44 to both sides of the equation:

12+4=4x−4+412 + 4 = 4x - 4 + 4

This simplifies to:

16=4x16 = 4x

Next, we can divide both sides of the equation by 44 to solve for xx:

164=4x4\frac{16}{4} = \frac{4x}{4}

This simplifies to:

4=x4 = x

Evaluating the Possible Solutions

Now that we have solved for xx, let's evaluate the possible solutions:

  • A. x=52x = \frac{5}{2}: This is not a possible solution, since we found that x=4x = 4.
  • B. x=2x = 2: This is not a possible solution, since we found that x=4x = 4.
  • C. x=log⁡9x = \log 9: This is not a possible solution, since we found that x=4x = 4.
  • D. x=4x = 4: This is the correct solution, since we found that x=4x = 4.

Conclusion

In this article, we solved an exponential equation, 96=(32)2x−29^6 = \left(3^2\right)^{2x-2}, and found that the possible solution is x=4x = 4. We used the properties of exponents to simplify the equation and solve for xx. This problem requires a deep understanding of algebraic manipulations and properties of exponents, making it an excellent example of a challenging exponential equation.

Additional Tips and Tricks

When solving exponential equations, it's essential to remember the following tips and tricks:

  • Use the properties of exponents to simplify the equation.
  • Equate the exponents when the bases are the same.
  • Solve for the variable by adding, subtracting, multiplying, or dividing both sides of the equation by the same value.
  • Check the possible solutions by plugging them back into the original equation.

By following these tips and tricks, you can become proficient in solving exponential equations and tackle even the most challenging problems with confidence.

Common Mistakes to Avoid

When solving exponential equations, it's essential to avoid the following common mistakes:

  • Failing to simplify the equation using the properties of exponents.
  • Not equating the exponents when the bases are the same.
  • Not solving for the variable by adding, subtracting, multiplying, or dividing both sides of the equation by the same value.
  • Not checking the possible solutions by plugging them back into the original equation.

By avoiding these common mistakes, you can ensure that your solutions are accurate and reliable.

Real-World Applications

Exponential equations have numerous real-world applications, including:

  • Modeling population growth and decline.
  • Calculating compound interest and investment returns.
  • Analyzing the spread of diseases and epidemics.
  • Understanding the behavior of complex systems and networks.

By mastering the art of solving exponential equations, you can unlock a wide range of applications and make a significant impact in various fields.

Conclusion

Introduction

In our previous article, we explored the concept of solving exponential equations and provided a step-by-step guide on how to solve a specific equation, 96=(32)2x−29^6 = \left(3^2\right)^{2x-2}. In this article, we will address some of the most frequently asked questions related to solving exponential equations.

Q&A

Q: What is an exponential equation?

A: An exponential equation is an equation that involves an exponential expression, which is a number raised to a power. For example, aba^b is an exponential expression, where aa is the base and bb is the exponent.

Q: How do I simplify an exponential equation?

A: To simplify an exponential equation, you can use the properties of exponents, such as:

  • (am)n=amn(a^m)^n = a^{mn}
  • am⋅an=am+na^m \cdot a^n = a^{m+n}
  • aman=am−n\frac{a^m}{a^n} = a^{m-n}

Q: How do I solve for x in an exponential equation?

A: To solve for x in an exponential equation, you can use the following steps:

  1. Simplify the equation using the properties of exponents.
  2. Equate the exponents when the bases are the same.
  3. Solve for x by adding, subtracting, multiplying, or dividing both sides of the equation by the same value.

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 using the properties of exponents.
  • Not equating the exponents when the bases are the same.
  • Not solving for the variable by adding, subtracting, multiplying, or dividing both sides of the equation by the same value.
  • Not checking the possible solutions by plugging them back into the original equation.

Q: How do I check my solutions?

A: To check your solutions, you can plug them back into the original equation and verify that they are true. This will help you ensure that your solutions are accurate and reliable.

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 and investment returns.
  • Analyzing the spread of diseases and epidemics.
  • Understanding the behavior of complex systems and networks.

Q: How can I practice solving exponential equations?

A: To practice solving exponential equations, you can try the following:

  • Start with simple equations and gradually move on to more complex ones.
  • Use online resources, such as practice problems and worksheets.
  • Join a study group or find a study partner to work on problems together.
  • Review and practice regularly to build your skills and confidence.

Conclusion

In conclusion, solving exponential equations requires a deep understanding of algebraic manipulations and properties of exponents. By following the tips and tricks outlined in this article, you can become proficient in solving exponential equations and tackle even the most challenging problems with confidence. Remember to avoid common mistakes and explore the real-world applications of exponential equations to make a significant impact in various fields.

Additional Resources

For further practice and review, we recommend the following resources:

  • Khan Academy: Exponents and Exponential Functions
  • Mathway: Exponential Equations
  • IXL: Exponents and Exponential Functions
  • MIT OpenCourseWare: Exponents and Exponential Functions

By utilizing these resources, you can further develop your skills and knowledge in solving exponential equations and become proficient in this critical area of mathematics.