Select The Correct Answer.What Is The Solution To This Equation? 216 = 6 7 M − 1 216=6^{7m-1} 216 = 6 7 M − 1 A. X = 2.5 X=2.5 X = 2.5 B. X = 2 X=2 X = 2 C. X = 1 X=1 X = 1 D. Z = 3 Z=3 Z = 3

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 the equation $216=6^{7m-1}$, and we will explore the different methods and techniques used to find the solution.

Understanding Exponential Equations

Exponential equations are equations that involve an exponential expression, which is an expression of the form $a^b$, where $a$ is the base and $b$ is the exponent. In the equation $216=6^{7m-1}$, the base is $6$ and the exponent is $7m-1$.

The Properties of Exponents

Before we can solve the equation, we need to understand the properties of exponents. The properties of exponents are as follows:

• Product of Powers: When we multiply two exponential expressions with the same base, we add the exponents. For example, $a^m \cdot a^n = a^{m+n}$.
• Power of a Power: When we raise an exponential expression to a power, we multiply the exponents. For example, $(a^m)^n = a^{mn}$.
• Zero Exponent: Any non-zero number raised to the power of zero is equal to 1. For example, $a^0 = 1$.
• Negative Exponent: Any non-zero number raised to a negative power is equal to the reciprocal of the number raised to the positive power. For example, $a^{-m} = \frac{1}{a^m}$.

Solving the Equation

Now that we have a good understanding of the properties of exponents, we can solve the equation $216=6^{7m-1}$.

Step 1: Rewrite the Equation

The first step in solving the equation is to rewrite it in a more manageable form. We can do this by expressing $216$ as a power of $6$. Since $216 = 6^3$, we can rewrite the equation as:

$$6^3 = 6^{7m-1}$$

Step 2: Equate the Exponents

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

$$3 = 7m-1$$

Step 3: Solve for m

Now that we have a linear equation, we can solve for $m$:

$$3 + 1 = 7m$$

$$4 = 7m$$

$$m = \frac{4}{7}$$

Step 4: Check the Solution

To check our solution, we can plug it back into the original equation:

$$216 = 6^{7\left(\frac{4}{7}\right)-1}$$

$$216 = 6^{4-1}$$

$$216 = 6^3$$

Since the equation holds true, we have found the correct solution.

Conclusion

In this article, we have solved the equation $216=6^{7m-1}$ using the properties of exponents. We have shown that the solution is $m = \frac{4}{7}$. We have also checked our solution by plugging it back into the original equation.

The Final Answer

The final answer is $\boxed{\frac{4}{7}}$.

Discussion

What do you think about the solution to this equation? Do you have any questions or comments about the steps involved in solving it? Share your thoughts in the discussion section below.

Discussion Section

• Question 1: How do you think we can apply the properties of exponents to solve other exponential equations?
• Question 2: Can you think of any other methods for solving exponential equations?
• Comment 1: I found the solution to be very straightforward. I like how you broke down the steps involved in solving the equation.
• Comment 2: I had a hard time understanding the properties of exponents. Can you explain them in more detail?

Answer to Question 1

The properties of exponents can be applied to solve other exponential equations by using the following methods:

• Method 1: Rewrite the equation in a more manageable form by expressing the base as a power of a smaller base.
• Method 2: Use the product of powers property to combine the exponents.
• Method 3: Use the power of a power property to simplify the exponent.

Answer to Question 2

Yes, there are other methods for solving exponential equations, including:

• Method 1: Using logarithms to solve the equation.
• Method 2: Using the change of base formula to solve the equation.
• Method 3: Using the properties of exponents to simplify the equation.

Answer to Comment 1

I'm glad you found the solution to be straightforward. I tried to break down the steps involved in solving the equation in a clear and concise manner. If you have any questions or need further clarification, feel free to ask.

Answer to Comment 2

I'd be happy to explain the properties of exponents in more detail. The properties of exponents are as follows:

• Product of Powers: When we multiply two exponential expressions with the same base, we add the exponents. For example, $a^m \cdot a^n = a^{m+n}$.
• Power of a Power: When we raise an exponential expression to a power, we multiply the exponents. For example, $(a^m)^n = a^{mn}$.
• Zero Exponent: Any non-zero number raised to the power of zero is equal to 1. For example, $a^0 = 1$.
• Negative Exponent: Any non-zero number raised to a negative power is equal to the reciprocal of the number raised to the positive power. For example, $a^{-m} = \frac{1}{a^m}$.

Introduction

In our previous article, we solved the equation $216=6^{7m-1}$ using the properties of exponents. In this article, we will answer some of the most frequently asked questions about solving exponential equations.

Q: What are the properties of exponents?

A: The properties of exponents are as follows:

• Product of Powers: When we multiply two exponential expressions with the same base, we add the exponents. For example, $a^m \cdot a^n = a^{m+n}$.
• Power of a Power: When we raise an exponential expression to a power, we multiply the exponents. For example, $(a^m)^n = a^{mn}$.
• Zero Exponent: Any non-zero number raised to the power of zero is equal to 1. For example, $a^0 = 1$.
• Negative Exponent: Any non-zero number raised to a negative power is equal to the reciprocal of the number raised to the positive power. For example, $a^{-m} = \frac{1}{a^m}$.

Q: How do I apply the properties of exponents to solve exponential equations?

A: To apply the properties of exponents to solve exponential equations, follow these steps:

1. Rewrite the equation: Rewrite the equation in a more manageable form by expressing the base as a power of a smaller base.
2. Use the product of powers property: Use the product of powers property to combine the exponents.
3. Use the power of a power property: Use the power of a power property to simplify the exponent.
4. Solve for the variable: Solve for the variable by isolating it on one side of the equation.

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

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

• Not rewriting the equation: Failing to rewrite the equation in a more manageable form can make it difficult to apply the properties of exponents.
• Not using the product of powers property: Failing to use the product of powers property can make it difficult to combine the exponents.
• Not using the power of a power property: Failing to use the power of a power property can make it difficult to simplify the exponent.
• Not solving for the variable: Failing to solve for the variable can make it difficult to find the solution to the equation.

Q: How do I check my solution to an exponential equation?

A: To check your solution to an exponential equation, follow these steps:

1. Plug the solution back into the original equation: Plug the solution back into the original equation to see if it holds true.
2. Simplify the equation: Simplify the equation to see if it is equal to the original equation.
3. Check the solution: Check the solution to see if it is correct.

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

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

• Population growth: Exponential equations can be used to model population growth and decline.
• Financial modeling: Exponential equations can be used to model financial growth and decline.
• Science and engineering: Exponential equations can be used to model scientific and engineering phenomena, such as the growth of bacteria and the decay of radioactive materials.

Conclusion

In this article, we have answered some of the most frequently asked questions about solving exponential equations. We have also provided some tips and tricks for solving exponential equations and checking your solution. By following these tips and tricks, you can become a master of solving exponential equations and apply them to real-world problems.

Additional Resources

For more information on solving exponential equations, check out the following resources:

• Online tutorials: There are many online tutorials available that can help you learn how to solve exponential equations.
• Textbooks: There are many textbooks available that can help you learn how to solve exponential equations.
• Practice problems: There are many practice problems available that can help you practice solving exponential equations.

Discussion

What do you think about solving exponential equations? Do you have any questions or comments about the tips and tricks provided in this article? Share your thoughts in the discussion section below.

Discussion Section

• Question 1: How do you think exponential equations can be applied to real-world problems?
• Question 2: Can you think of any other tips and tricks for solving exponential equations?
• Comment 1: I found the tips and tricks provided in this article to be very helpful. I like how you broke down the steps involved in solving exponential equations.
• Comment 2: I had a hard time understanding the properties of exponents. Can you explain them in more detail?

Answer to Question 1

Exponential equations can be applied to real-world problems in many ways, including:

• Modeling population growth: Exponential equations can be used to model population growth and decline.
• Financial modeling: Exponential equations can be used to model financial growth and decline.
• Science and engineering: Exponential equations can be used to model scientific and engineering phenomena, such as the growth of bacteria and the decay of radioactive materials.

Answer to Question 2

Yes, there are many other tips and tricks for solving exponential equations, including:

• Using logarithms: Logarithms can be used to simplify exponential equations and make them easier to solve.
• Using the change of base formula: The change of base formula can be used to simplify exponential equations and make them easier to solve.
• Using the properties of exponents: The properties of exponents can be used to simplify exponential equations and make them easier to solve.

Answer to Comment 1

I'm glad you found the tips and tricks provided in this article to be helpful. I tried to break down the steps involved in solving exponential equations in a clear and concise manner. If you have any questions or need further clarification, feel free to ask.

Answer to Comment 2

I'd be happy to explain the properties of exponents in more detail. The properties of exponents are as follows:

• Product of Powers: When we multiply two exponential expressions with the same base, we add the exponents. For example, $a^m \cdot a^n = a^{m+n}$.
• Power of a Power: When we raise an exponential expression to a power, we multiply the exponents. For example, $(a^m)^n = a^{mn}$.
• Zero Exponent: Any non-zero number raised to the power of zero is equal to 1. For example, $a^0 = 1$.
• Negative Exponent: Any non-zero number raised to a negative power is equal to the reciprocal of the number raised to the positive power. For example, $a^{-m} = \frac{1}{a^m}$.

I hope this helps clarify the properties of exponents. If you have any further questions, feel free to ask.