For What Value Of $a$ Does $\left(\frac{1}{7}\right)^{3a+3}=343^{2-1}$?A. -1 B. 0 C. 1 D. No Solution

Introduction

In this article, we will delve into solving for the value of $a$ in the given exponential equation $\left(\frac{1}{7}\right)^{3a+3}=343^{2-1}$. This equation involves exponentiation and requires us to manipulate the expressions to isolate the variable $a$. We will use properties of exponents, including the power rule and the product rule, to simplify the equation and solve for $a$.

Understanding the Equation

The given equation is $\left(\frac{1}{7}\right)^{3a+3}=343^{2-1}$. To begin solving this equation, we need to simplify the expressions on both sides. We can rewrite $343$ as $7^3$, since $343$ is equal to $7$ cubed. This gives us $\left(\frac{1}{7}\right)^{3a+3}=(7^3)^{2-1}$.

Simplifying the Right-Hand Side

Using the property of exponents that $(a^m)^n=a^{mn}$, we can simplify the right-hand side of the equation. This gives us $\left(\frac{1}{7}\right)^{3a+3}=7^{3(2-1)}$. Simplifying further, we get $\left(\frac{1}{7}\right)^{3a+3}=7^3$.

Equating the Exponents

Since the bases on both sides of the equation are the same, we can equate the exponents. This gives us the equation $3a+3=3$. We can now solve for $a$ by isolating the variable.

Solving for $a$

To solve for $a$, we need to isolate the variable on one side of the equation. Subtracting $3$ from both sides gives us $3a=0$. Dividing both sides by $3$ gives us $a=0$.

Conclusion

In this article, we solved for the value of $a$ in the given exponential equation $\left(\frac{1}{7}\right)^{3a+3}=343^{2-1}$. We simplified the expressions on both sides of the equation, equated the exponents, and solved for $a$. The value of $a$ that satisfies the equation is $a=0$.

Final Answer

The final answer is $\boxed{0}$.

Step-by-Step Solution

Here are the step-by-step steps to solve the equation:

1. Simplify the right-hand side of the equation: $\left(\frac{1}{7}\right)^{3a+3}=(7^3)^{2-1}$
2. Use the property of exponents to simplify the right-hand side: $\left(\frac{1}{7}\right)^{3a+3}=7^{3(2-1)}$
3. Simplify further: $\left(\frac{1}{7}\right)^{3a+3}=7^3$
4. Equate the exponents: $3a+3=3$
5. Subtract $3$ from both sides: $3a=0$
6. Divide both sides by $3$: $a=0$

Common Mistakes to Avoid

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

• Always simplify the expressions on both sides of the equation.
• Use the properties of exponents to simplify the expressions.
• Equate the exponents when the bases are the same.
• Solve for the variable by isolating it on one side of the equation.

By following these steps and avoiding common mistakes, you can confidently solve exponential equations like the one in this article.

Introduction

In the previous article, we solved for the value of $a$ in the given exponential equation $\left(\frac{1}{7}\right)^{3a+3}=343^{2-1}$. In this article, we will address some frequently asked questions (FAQs) related to solving exponential equations. We will provide detailed answers to common questions and provide additional tips and resources for further learning.

Q: What is the first step in solving an exponential equation?

A: The first step in solving an exponential equation is to simplify the expressions on both sides of the equation. This involves using the properties of exponents, such as the power rule and the product rule, to rewrite the expressions in a more manageable form.

Q: How do I know when to use the power rule and the product rule?

A: The power rule is used when you have an expression in the form $(a^m)^n=a^{mn}$. The product rule is used when you have an expression in the form $a^m \cdot a^n=a^{m+n}$. You can use the power rule and the product rule to simplify expressions and make them easier to work with.

Q: What if the bases on both sides of the equation are not the same?

A: If the bases on both sides of the equation are not the same, you cannot equate the exponents. In this case, you need to use logarithms to solve the equation. Logarithms allow you to work with equations that have different bases.

Q: How do I use logarithms to solve an exponential equation?

A: To use logarithms to solve an exponential equation, you need to take the logarithm of both sides of the equation. This will allow you to use the properties of logarithms to simplify the equation and solve for the variable.

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

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

• Not simplifying the expressions on both sides of the equation
• Not using the properties of exponents correctly
• Not equating the exponents when the bases are the same
• Not using logarithms when the bases are not the same

Q: Where can I find more resources on solving exponential equations?

A: There are many resources available online and in textbooks that can help you learn how to solve exponential equations. Some popular resources include:

• Khan Academy: Khan Academy has a comprehensive section on exponential equations that includes video lessons and practice exercises.
• Mathway: Mathway is an online math problem solver that can help you solve exponential equations and other types of math problems.
• Wolfram Alpha: Wolfram Alpha is a powerful online calculator that can help you solve exponential equations and other types of math problems.

Conclusion

In this article, we addressed some frequently asked questions (FAQs) related to solving exponential equations. We provided detailed answers to common questions and provided additional tips and resources for further learning. By following the steps and tips outlined in this article, you can confidently solve exponential equations and improve your math skills.

Final Tips

• Always simplify the expressions on both sides of the equation.
• Use the properties of exponents correctly.
• Equate the exponents when the bases are the same.
• Use logarithms when the bases are not the same.
• Practice, practice, practice! The more you practice solving exponential equations, the more confident you will become.

Additional Resources

• Khan Academy: Exponential Equations
• Mathway: Exponential Equations
• Wolfram Alpha: Exponential Equations
• MIT OpenCourseWare: Exponential Equations
• Purplemath: Exponential Equations