Using the Property of Exponents

Oct 13, 2025 by miodeya.com 112 views

Introduction

In this article, we will delve into solving for the value of $n$ in the given equation $\frac{216^{n-2}}{\left(\frac{1}{36}\right)^{3n}}=216$ . This equation involves exponentiation and fractions, making it a challenging problem to solve. We will break down the solution step by step, using algebraic manipulations and properties of exponents to simplify the equation and isolate the variable $n$ .

Simplifying the Equation

The first step in solving the equation is to simplify the expression on the left-hand side. We can start by rewriting the fraction as a product of two terms:

\frac{216^{n-2}}{\left(\frac{1}{36}\right)^{3n}} = 216^{n-2} \cdot \left(36\right)^{3n}

Next, we can simplify the expression inside the parentheses by rewriting $36$ as $6^2$ :

216^{n-2} \cdot \left(36\right)^{3n} = 216^{n-2} \cdot \left(6^2\right)^{3n}

Using the property of exponents that states $\left(a^b\right)^c = a^{bc}$ , we can simplify the expression further:

216^{n-2} \cdot \left(6^2\right)^{3n} = 216^{n-2} \cdot 6^{6n}

Equating the Exponents

Now that we have simplified the expression, we can equate the exponents on both sides of the equation:

216^{n-2} \cdot 6^{6n} = 216

Since $216 = 6^3$ , we can rewrite the equation as:

216^{n-2} \cdot 6^{6n} = 6^3

Using the Property of Exponents

Using the property of exponents that states $a^b \cdot a^c = a^{b+c}$ , we can simplify the expression on the left-hand side:

216^{n-2} \cdot 6^{6n} = 6^{3+6n}

Equating the Exponents Again

Now that we have simplified the expression, we can equate the exponents on both sides of the equation:

6^{3+6n} = 6^3

Solving for $n$

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

3+6n = 3

Subtracting $3$ from both sides gives us:

6n = 0

Dividing both sides by $6$ gives us:

n = 0

However, this is not the only possible solution. We can also divide both sides by $6$ and then multiply both sides by $-1$ to get:

n = -\frac{1}{3}

Conclusion

In this article, we have solved for the value of $n$ in the given equation $\frac{216^{n-2}}{\left(\frac{1}{36}\right)^{3n}}=216$ . We have broken down the solution step by step, using algebraic manipulations and properties of exponents to simplify the equation and isolate the variable $n$ . The final answer is $n = -\frac{1}{3}$ .

Final Answer

The final answer is $\boxed{-\frac{1}{3}}$ .

Introduction

In our previous article, we solved for the value of $n$ in the given equation $\frac{216^{n-2}}{\left(\frac{1}{36}\right)^{3n}}=216$ . However, we received many questions from readers who were struggling to understand the solution. In this article, we will address some of the most frequently asked questions (FAQs) about solving for the value of $n$ .

Q: What is the main concept behind solving for the value of $n$ ?

A: The main concept behind solving for the value of $n$ is to use algebraic manipulations and properties of exponents to simplify the equation and isolate the variable $n$ .

Q: Why do we need to simplify the expression on the left-hand side of the equation?

A: We need to simplify the expression on the left-hand side of the equation to make it easier to work with and to isolate the variable $n$ . By simplifying the expression, we can use properties of exponents to rewrite the equation in a more manageable form.

Q: What is the property of exponents that we used to simplify the expression?

A: The property of exponents that we used to simplify the expression is $\left(a^b\right)^c = a^{bc}$ . This property allows us to rewrite the expression inside the parentheses as a single exponent.

Q: Why do we need to equate the exponents on both sides of the equation?

A: We need to equate the exponents on both sides of the equation because the bases are the same. By equating the exponents, we can solve for the value of $n$ .

Q: What is the final answer to the problem?

A: The final answer to the problem is $n = -\frac{1}{3}$ .

Q: Can you explain why we got two possible solutions for the value of $n$ ?

A: Yes, we got two possible solutions for the value of $n$ because we divided both sides of the equation by $6$ and then multiplied both sides by $-1$ . This resulted in two possible values for $n$ : $n = 0$ and $n = -\frac{1}{3}$ .

Q: Which solution is correct?

A: The correct solution is $n = -\frac{1}{3}$ . The solution $n = 0$ is not valid because it does not satisfy the original equation.

Q: Can you provide more examples of solving for the value of $n$ ?

A: Yes, we can provide more examples of solving for the value of $n$ . However, the examples will depend on the specific equation and the properties of exponents that are used.

Q: How can I apply the concepts learned in this article to other problems?

A: You can apply the concepts learned in this article to other problems by using algebraic manipulations and properties of exponents to simplify the equation and isolate the variable. Additionally, you can use the property of exponents that we used to simplify the expression to rewrite the equation in a more manageable form.

Conclusion

In this article, we have addressed some of the most frequently asked questions (FAQs) about solving for the value of $n$ . We have provided explanations and examples to help readers understand the concepts and apply them to other problems. By following the steps outlined in this article, readers can develop a deeper understanding of solving for the value of $n$ and apply the concepts to other problems.

Final Answer

The final answer is $\boxed{-\frac{1}{3}}$ .