Solve The Equation: $\[ 243^{-y} = \left(\frac{1}{243}\right)^{3y} \cdot 9^{-2y} \\]Possible Solutions:A. \[$ Y = -1 \$\]B. \[$ Y = 0 \$\]C. \[$ Y = 1 \$\]D. No Solution

Introduction

In this article, we will delve into solving a complex equation involving exponents and fractions. The equation is given as: ${ 243^{-y} = \left(\frac{1}{243}\right)^{3y} \cdot 9^{-2y} }$. Our goal is to find the possible solutions for the variable $y$. We will break down the solution into manageable steps, making it easier to understand and follow.

Step 1: Simplify the Equation

To start solving the equation, we need to simplify it by expressing both sides using the same base. We can rewrite $243$ as $3^5$ and $\frac{1}{243}$ as $3^{-5}$. Similarly, we can express $9$ as $3^2$. Substituting these values into the equation, we get:

$${ (3^5)^{-y} = (3^{-5})^{3y} \cdot (3^2)^{-2y} }$$

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

$${ 3^{-5y} = 3^{-15y} \cdot 3^{-4y} }$$

Step 2: Combine Like Terms

Now that we have the same base on both sides, we can combine like terms. We can rewrite the equation as:

$${ 3^{-5y} = 3^{-15y - 4y} }$$

Simplifying the exponent on the right-hand side, we get:

$${ 3^{-5y} = 3^{-19y} }$$

Step 3: Equate Exponents

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

$${ -5y = -19y }$$

Step 4: Solve for $y$

Now that we have a simple equation, we can solve for $y$. Adding $19y$ to both sides, we get:

$${ 14y = 0 }$$

Dividing both sides by $14$, we get:

$${ y = 0 }$$

Conclusion

In this article, we solved the equation ${ 243^{-y} = \left(\frac{1}{243}\right)^{3y} \cdot 9^{-2y} }$. By simplifying the equation, combining like terms, and equating exponents, we found that the possible solution for the variable $y$ is $y = 0$. This solution satisfies the original equation, making it the correct answer.

Discussion

The equation we solved is a classic example of an exponential equation. It involves fractions and negative exponents, making it a challenging problem to solve. However, by breaking down the solution into manageable steps, we were able to find the correct answer. This problem requires a good understanding of exponents, fractions, and algebraic manipulations.

Possible Solutions

Based on our solution, we can conclude that the possible solution for the variable $y$ is:

• A. $y = -1$: This solution does not satisfy the original equation.
• B. $y = 0$: This solution satisfies the original equation and is the correct answer.
• C. $y = 1$: This solution does not satisfy the original equation.
• D. No solution: This option is incorrect, as we found a valid solution for $y$.

Final Answer

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Introduction

In our previous article, we solved the equation ${ 243^{-y} = \left(\frac{1}{243}\right)^{3y} \cdot 9^{-2y} }$. We found that the possible solution for the variable $y$ is $y = 0$. In this article, we will provide a Q&A guide to help you understand the solution and answer any questions you may have.

Q: What is the main concept behind solving this equation?

A: The main concept behind solving this equation is to simplify it by expressing both sides using the same base. We can rewrite $243$ as $3^5$ and $\frac{1}{243}$ as $3^{-5}$. Similarly, we can express $9$ as $3^2$. By using these properties, we can simplify the equation and find the solution.

Q: Why did we need to combine like terms?

A: We needed to combine like terms to simplify the equation further. By combining like terms, we can rewrite the equation as ${ 3^{-5y} = 3^{-19y} }$. This simplification helps us to equate the exponents and find the solution.

Q: How did we equate the exponents?

A: We equated the exponents by setting the two expressions equal to each other: ${ -5y = -19y }$. This step is crucial in solving the equation, as it allows us to find the value of $y$.

Q: Why did we add $19y$ to both sides?

A: We added $19y$ to both sides to isolate the variable $y$. By doing so, we can solve for $y$ and find the solution.

Q: What is the final answer?

A: The final answer is $\boxed{0}$. This solution satisfies the original equation and is the correct answer.

Q: What are some common mistakes to avoid when solving this equation?

A: Some common mistakes to avoid when solving this equation include:

• Not simplifying the equation by expressing both sides using the same base.
• Not combining like terms to simplify the equation further.
• Not equating the exponents correctly.
• Not solving for $y$ correctly.

Q: How can I apply this concept to other problems?

A: You can apply this concept to other problems by simplifying the equation by expressing both sides using the same base. Then, combine like terms to simplify the equation further. Finally, equate the exponents and solve for the variable.

Q: What are some tips for solving exponential equations?

A: Some tips for solving exponential equations include:

• Simplifying the equation by expressing both sides using the same base.
• Combining like terms to simplify the equation further.
• Equating the exponents correctly.
• Solving for the variable correctly.

Conclusion

In this article, we provided a Q&A guide to help you understand the solution to the equation ${ 243^{-y} = \left(\frac{1}{243}\right)^{3y} \cdot 9^{-2y} }$. We hope that this guide has been helpful in answering your questions and providing a better understanding of the solution.