For What Value Of $n$ Does ( 1 36 ) N = 216 \left(\frac{1}{36}\right)^n=216 ( 36 1 ​ ) N = 216 ?A. − 3 -3 − 3 B. − 3 2 -\frac{3}{2} − 2 3 ​ C. 3 2 \frac{3}{2} 2 3 ​ D. 3 3 3

$$

Introduction

In this article, we will delve into solving for the value of $n$ in the equation $\left(\frac{1}{36}\right)^n=216$. This equation involves exponentiation and requires us to manipulate the equation to isolate the variable $n$. We will use algebraic techniques to solve for $n$ and provide a step-by-step solution.

Understanding the Equation

The given equation is $\left(\frac{1}{36}\right)^n=216$. To begin solving for $n$, we need to understand the properties of exponents and how to manipulate the equation to isolate the variable.

Simplifying the Equation

We can start by simplifying the left-hand side of the equation. We know that $\frac{1}{36}=\left(\frac{1}{6}\right)^2$. Therefore, we can rewrite the equation as $\left(\left(\frac{1}{6}\right)^2\right)^n=216$.

Using Exponent Rules

Using the rule of exponents that states $\left(a^m\right)^n=a^{mn}$, we can simplify the left-hand side of the equation further. This gives us $\left(\frac{1}{6}\right)^{2n}=216$.

Expressing 216 as a Power of 6

We can rewrite 216 as a power of 6. Since $6^3=216$, we can express 216 as $6^3$. Therefore, we can rewrite the equation as $\left(\frac{1}{6}\right)^{2n}=6^3$.

Using Exponent Rules Again

Using the rule of exponents that states $\left(\frac{1}{a}\right)^n=a^{-n}$, we can rewrite the left-hand side of the equation as $\left(6^2\right)^{-n}=6^3$.

Simplifying the Equation Further

Using the rule of exponents that states $\left(a^m\right)^n=a^{mn}$, we can simplify the left-hand side of the equation further. This gives us $6^{-2n}=6^3$.

Equating Exponents

Since the bases are the same, we can equate the exponents. This gives us $-2n=3$.

Solving for n

To solve for $n$, we can divide both sides of the equation by $-2$. This gives us $n=-\frac{3}{2}$.

Conclusion

In this article, we solved for the value of $n$ in the equation $\left(\frac{1}{36}\right)^n=216$. We used algebraic techniques to manipulate the equation and isolate the variable $n$. The final solution is $n=-\frac{3}{2}$.

Final Answer

The final answer is $\boxed{-\frac{3}{2}}$.

Discussion

The solution to this problem involves using exponent rules and manipulating the equation to isolate the variable $n$. The key steps in solving this problem are:

• Simplifying the left-hand side of the equation using exponent rules
• Expressing 216 as a power of 6
• Using exponent rules to simplify the equation further
• Equating exponents since the bases are the same
• Solving for $n$ by dividing both sides of the equation by $-2$

This problem requires a good understanding of exponent rules and algebraic techniques. With practice and experience, solving equations involving exponents becomes more manageable.

Related Problems

If you are interested in practicing more problems involving exponents, here are a few related problems:

• Solve for $n$ in the equation $\left(\frac{1}{4}\right)^n=16$
• Solve for $n$ in the equation $\left(\frac{1}{9}\right)^n=81$
• Solve for $n$ in the equation $\left(\frac{1}{25}\right)^n=625$

These problems require similar techniques and algebraic manipulations as the original problem. With practice, you can become more confident and proficient in solving equations involving exponents.

$$

Introduction

In our previous article, we solved for the value of $n$ in the equation $\left(\frac{1}{36}\right)^n=216$. We used algebraic techniques to manipulate the equation and isolate the variable $n$. In this article, we will provide a Q&A section to help clarify any doubts and provide additional insights into solving this type of equation.

Q: What is the first step in solving the equation $\left(\frac{1}{36}\right)^n=216$?

A: The first step is to simplify the left-hand side of the equation by expressing $\frac{1}{36}$ as a power of 6. This gives us $\left(\left(\frac{1}{6}\right)^2\right)^n=216$.

Q: How do we simplify the left-hand side of the equation further?

A: We can use the rule of exponents that states $\left(a^m\right)^n=a^{mn}$. This gives us $\left(\frac{1}{6}\right)^{2n}=216$.

Q: How do we express 216 as a power of 6?

A: We can rewrite 216 as $6^3$. Therefore, we can express 216 as $6^3$.

Q: How do we simplify the equation further?

A: We can use the rule of exponents that states $\left(\frac{1}{a}\right)^n=a^{-n}$. This gives us $\left(6^2\right)^{-n}=6^3$.

Q: How do we simplify the left-hand side of the equation further?

A: We can use the rule of exponents that states $\left(a^m\right)^n=a^{mn}$. This gives us $6^{-2n}=6^3$.

Q: How do we equate the exponents?

A: Since the bases are the same, we can equate the exponents. This gives us $-2n=3$.

Q: How do we solve for n?

A: To solve for $n$, we can divide both sides of the equation by $-2$. This gives us $n=-\frac{3}{2}$.

Q: What is the final answer?

A: The final answer is $n=-\frac{3}{2}$.

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

A: Some common mistakes to avoid include:

• Not simplifying the left-hand side of the equation properly
• Not expressing 216 as a power of 6
• Not using the correct exponent rules
• Not equating the exponents properly
• Not solving for $n$ correctly

Q: What are some tips for solving this type of equation?

A: Some tips for solving this type of equation include:

• Simplifying the left-hand side of the equation as much as possible
• Expressing 216 as a power of 6
• Using the correct exponent rules
• Equating the exponents properly
• Solving for $n$ correctly

Q: Can you provide some additional practice problems?

A: Yes, here are a few additional practice problems:

• Solve for $n$ in the equation $\left(\frac{1}{4}\right)^n=16$
• Solve for $n$ in the equation $\left(\frac{1}{9}\right)^n=81$
• Solve for $n$ in the equation $\left(\frac{1}{25}\right)^n=625$

These problems require similar techniques and algebraic manipulations as the original problem. With practice, you can become more confident and proficient in solving equations involving exponents.

Conclusion

In this Q&A article, we provided additional insights and clarification on solving the equation $\left(\frac{1}{36}\right)^n=216$. We covered common mistakes to avoid and tips for solving this type of equation. We also provided additional practice problems to help you become more confident and proficient in solving equations involving exponents.