Solve For { N $}$ In The Equation: ${ 6^{-2} \times 6^n = 1 }$

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Introduction

In mathematics, solving equations is a fundamental concept that helps us understand various mathematical operations and relationships. One of the essential skills in solving equations is to isolate the variable, which in this case is ${ n }$. In this article, we will focus on solving for ${ n }$ in the equation ${ 6^{-2} \times 6^n = 1 }$. We will break down the solution step by step, making it easy to understand and follow.

Understanding the Equation

The given equation is ${ 6^{-2} \times 6^n = 1 }$. To solve for ${ n }$, we need to isolate the variable ${ n }$ on one side of the equation. The equation involves exponentiation, which can be simplified using the properties of exponents.

Simplifying the Equation

To simplify the equation, we can start by rewriting ${ 6^{-2} }$ as ${ \frac{1}{6^2} }$. This gives us:

${ \frac{1}{6^2} \times 6^n = 1 }$

Using Exponent Properties

We can simplify the equation further by using the property of exponents that states ${ a^m \times a^n = a^{m+n} }$. Applying this property to the equation, we get:

${ 6^{-2} \times 6^n = 6^{-2+n} }$

Setting Up the Equation

Now that we have simplified the equation, we can set it up to solve for ${ n }$. We know that ${ 6^{-2+n} = 1 }$, which means that the exponent must be equal to zero. Therefore, we can set up the equation as follows:

${ -2 + n = 0 }$

Solving for ${ n }$

To solve for ${ n }$, we can add 2 to both sides of the equation, which gives us:

${ n = 2 }$

Conclusion

In this article, we solved for ${ n }$ in the equation ${ 6^{-2} \times 6^n = 1 }$. We simplified the equation using exponent properties and set up the equation to solve for ${ n }$. The final solution is ${ n = 2 }$. This problem demonstrates the importance of understanding exponent properties and how to apply them to solve equations.

Additional Examples

Here are a few additional examples of solving for ${ n }$ in similar equations:

Example 1

Solve for ${ n }$ in the equation ${ 3^{-2} \times 3^n = 1 }$.

Solution

Using the same steps as before, we can simplify the equation and solve for ${ n }$:

${ 3^{-2} \times 3^n = 3^{-2+n} }$

Setting up the equation, we get:

${ -2 + n = 0 }$

Solving for ${ n }$, we get:

${ n = 2 }$

Example 2

Solve for ${ n }$ in the equation ${ 2^{-3} \times 2^n = 1 }$.

Solution

Using the same steps as before, we can simplify the equation and solve for ${ n }$:

${ 2^{-3} \times 2^n = 2^{-3+n} }$

Setting up the equation, we get:

${ -3 + n = 0 }$

Solving for ${ n }$, we get:

${ n = 3 }$

Final Thoughts

Solving for ${ n }$ in equations involving exponents requires a clear understanding of exponent properties and how to apply them to simplify the equation. By following the steps outlined in this article, you can solve for ${ n }$ in similar equations and develop a deeper understanding of mathematical operations and relationships.

References

• [1] Khan Academy. (n.d.). Exponents. Retrieved from https://www.khanacademy.org/math/algebra/x2f5f7d7/exponents
• [2] Mathway. (n.d.). Exponents. Retrieved from https://www.mathway.com/subjects/exponents

Related Topics

• [1] Solving Linear Equations
• [2] Solving Quadratic Equations
• [3] Exponent Properties

Tags

• [1] Exponents
• [2] Solving Equations
• [3] Algebra
• [4] Mathematics
  

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Introduction

In our previous article, we solved for ${ n }$ in the equation ${ 6^{-2} \times 6^n = 1 }$. However, we understand that sometimes, readers may have questions or need further clarification on the solution. In this Q&A article, we will address some common questions and provide additional insights on solving for ${ n }$ in similar equations.

Q: What is the significance of the exponent ${ -2 }$ in the equation?

A: The exponent ${ -2 }$ represents a negative exponent, which means that the base ${ 6 }$ is being raised to the power of ${ -2 }$. In this case, ${ 6^{-2} }$ is equivalent to ${ \frac{1}{6^2} }$, which simplifies to ${ \frac{1}{36} }$.

Q: How do I simplify the equation ${ 6^{-2} \times 6^n = 1 }$?

A: To simplify the equation, you can start by rewriting ${ 6^{-2} }$ as ${ \frac{1}{6^2} }$. This gives you:

${ \frac{1}{6^2} \times 6^n = 1 }$

Q: What is the property of exponents that allows us to simplify the equation?

A: The property of exponents that allows us to simplify the equation is ${ a^m \times a^n = a^{m+n} }$. Applying this property to the equation, we get:

${ 6^{-2} \times 6^n = 6^{-2+n} }$

Q: How do I set up the equation to solve for ${ n }$?

A: To set up the equation, you can equate the exponent ${ -2+n }$ to zero, since ${ 6^{-2+n} = 1 }$. This gives you:

${ -2 + n = 0 }$

Q: What is the final solution for ${ n }$?

A: Solving for ${ n }$, we get:

${ n = 2 }$

Q: Can I apply the same steps to solve for ${ n }$ in other equations?

A: Yes, you can apply the same steps to solve for ${ n }$ in other equations that involve exponents. The key is to simplify the equation using exponent properties and then set up the equation to solve for ${ n }$.

Q: What are some common mistakes to avoid when solving for ${ n }$?

A: Some common mistakes to avoid when solving for ${ n }$ include:

• Not simplifying the equation using exponent properties
• Not setting up the equation correctly to solve for ${ n }$
• Not checking the solution for validity

Q: How can I practice solving for ${ n }$ in equations?

A: You can practice solving for ${ n }$ in equations by working through examples and exercises. You can also try solving for ${ n }$ in different types of equations, such as linear and quadratic equations.

Q: What are some real-world applications of solving for ${ n }$ in equations?

A: Solving for ${ n }$ in equations has many real-world applications, including:

• Modeling population growth and decay
• Analyzing financial data and making predictions
• Solving optimization problems in business and economics

Conclusion

In this Q&A article, we addressed some common questions and provided additional insights on solving for ${ n }$ in the equation ${ 6^{-2} \times 6^n = 1 }$. We hope that this article has been helpful in clarifying any doubts and providing a deeper understanding of solving for ${ n }$ in equations.

Additional Resources

• [1] Khan Academy. (n.d.). Exponents. Retrieved from https://www.khanacademy.org/math/algebra/x2f5f7d7/exponents
• [2] Mathway. (n.d.). Exponents. Retrieved from https://www.mathway.com/subjects/exponents
• [3] Wolfram Alpha. (n.d.). Exponents. Retrieved from https://www.wolframalpha.com/input/?i=exponents

Tags

• [1] Exponents
• [2] Solving Equations
• [3] Algebra
• [4] Mathematics
• [5] Q&A