Solving Exponential Equations Finding The Value Of N In (1/36)^n=216

In this article, we will delve into solving the exponential equation $\left(\frac{1}{36}\right)^n = 216$. This problem is a classic example of how to manipulate exponents and bases to find an unknown variable. We'll break down each step, making it easy to understand even if you're new to exponential equations. Our focus will be on expressing both sides of the equation with a common base, which is a fundamental technique in solving such problems. We'll cover the basics of exponents, negative exponents, and fractional exponents, ensuring a comprehensive understanding. By the end of this discussion, you will not only be able to solve this specific problem but also tackle similar exponential equations with confidence. Let's embark on this mathematical journey together, making sure every step is clear and concise. The beauty of mathematics lies in its logical progression, and we'll follow that path meticulously to arrive at the correct answer. So, let's get started and unlock the secrets of exponential equations!

Understanding the Problem

To effectively solve the equation $\left(\frac{1}{36}\right)^n = 216$, it’s crucial to first understand the core concepts involved. This problem falls under the category of exponential equations, where the unknown variable, in this case, $n$, appears in the exponent. The key to solving such equations is to express both sides of the equation with the same base. This allows us to equate the exponents and solve for the unknown. In our case, we have $\frac{1}{36}$ on one side and $216$ on the other. We need to find a common base for both of these numbers. Recognizing the relationship between $36$ and $216$ is vital. Both are powers of $6$. Specifically, $36$ is $6^2$, and $216$ is $6^3$. Understanding this relationship is the first step towards simplifying the equation. The fraction $\frac{1}{36}$ can be written as $6^{-2}$, which is a crucial transformation. This step involves the concept of negative exponents, which we will explore in more detail. Furthermore, it's important to remember the properties of exponents, such as $(a^b)^c = a^{bc}$, which will be used to simplify the left side of the equation once we substitute $6^{-2}$ for $\frac{1}{36}$. By grasping these fundamental principles, we set the stage for a clear and methodical solution. Remember, the goal is not just to find the answer but to understand the underlying mathematical concepts that make the solution possible.

Expressing Both Sides with a Common Base

The heart of solving this exponential equation lies in expressing both sides with a common base. This technique allows us to directly compare the exponents and solve for the unknown variable $n$. As we identified earlier, both $\frac{1}{36}$ and $216$ can be expressed as powers of $6$. Let's break down how to do this systematically. First, consider $\frac{1}{36}$. We know that $36$ is $6^2$. Therefore, $\frac{1}{36}$ can be written as $\frac{1}{6^2}$. Now, recall the property of negative exponents: $a^{-b} = \frac{1}{a^b}$. Applying this property, we can rewrite $\frac{1}{6^2}$ as $6^{-2}$. So, $\frac{1}{36}$ is equivalent to $6^{-2}$. Next, let's look at $216$. We need to find a power of $6$ that equals $216$. By trying a few powers, we find that $6^1 = 6$, $6^2 = 36$, and $6^3 = 216$. Therefore, $216$ can be expressed as $6^3$. Now that we have expressed both $\frac{1}{36}$ and $216$ as powers of $6$, we can rewrite the original equation $\left(\frac{1}{36}\right)^n = 216$ as $(6^{-2})^n = 6^3$. This transformation is the crucial step in simplifying the equation. By expressing both sides with the same base, we pave the way for equating the exponents and solving for $n$. This method is a fundamental tool in dealing with exponential equations, and mastering it will greatly enhance your problem-solving skills in mathematics.

Simplifying the Equation

With both sides of the equation expressed using a common base, the next crucial step is simplifying the equation. We have transformed the original equation $\left(\frac{1}{36}\right)^n = 216$ into $(6^{-2})^n = 6^3$. Now, we need to apply the properties of exponents to further simplify the left side of the equation. Recall the power of a power rule: $(a^b)^c = a^{bc}$. This rule states that when you raise a power to another power, you multiply the exponents. Applying this rule to our equation, we have $(6^{-2})^n = 6^{-2n}$. So, the left side of the equation simplifies to $6^{-2n}$. Now our equation looks like this: $6^{-2n} = 6^3$. At this stage, the equation is significantly simpler than the original one. Both sides are expressed as powers of the same base, which allows us to directly compare the exponents. This simplification is a critical step in solving exponential equations. By reducing the equation to this form, we can now focus solely on the exponents and solve for $n$. This step highlights the importance of understanding and applying the properties of exponents. These properties are the tools that enable us to manipulate and simplify complex expressions, making them easier to solve. The ability to simplify equations is a fundamental skill in mathematics, and this example demonstrates how it can be effectively applied in the context of exponential equations.

Equating the Exponents

After simplifying the equation, we arrive at a pivotal point: equating the exponents. Our equation is now in the form $6^{-2n} = 6^3$. Since the bases on both sides of the equation are the same (both are $6$), we can equate the exponents. This is a fundamental principle in solving exponential equations: if $a^b = a^c$, then $b = c$. Applying this principle to our equation, we can set the exponents equal to each other: $-2n = 3$. This step transforms the exponential equation into a simple linear equation. By equating the exponents, we eliminate the exponential aspect of the problem and reduce it to a straightforward algebraic equation. This is a significant simplification that makes the equation much easier to solve. The ability to equate exponents is a direct consequence of the properties of exponential functions. It allows us to move from comparing exponential expressions to comparing their exponents, which are typically simpler to manipulate. This step is not only crucial for solving this particular problem but also a key technique for solving a wide range of exponential equations. By mastering this step, you gain a powerful tool for tackling mathematical problems involving exponents.

Solving for n

Having equated the exponents, we now face a simple linear equation: $-2n = 3$. The final step in solving the problem is to solve for n. This involves isolating $n$ on one side of the equation. To do this, we need to divide both sides of the equation by $-2$. Dividing both sides by $-2$, we get: $\frac{-2n}{-2} = \frac{3}{-2}$. This simplifies to: $n = -\frac{3}{2}$. Thus, we have found the value of $n$ that satisfies the original equation. This step is a straightforward application of basic algebraic principles. The key is to perform the same operation on both sides of the equation to maintain equality. In this case, dividing by $-2$ isolates $n$ and gives us the solution. The solution $n = -\frac{3}{2}$ means that when we substitute this value back into the original equation, $\left(\frac{1}{36}\right)^n = 216$, the equation holds true. This is a crucial step in verifying the solution. By finding the value of $n$, we have successfully solved the exponential equation. This process demonstrates the power of combining the properties of exponents with algebraic manipulation to solve mathematical problems. The ability to solve for unknowns is a fundamental skill in mathematics, and this example showcases how it can be applied in the context of exponential equations.

Verifying the Solution

To ensure the accuracy of our solution, it's essential to verify the solution. We found that $n = -\frac{3}{2}$. To verify this, we substitute this value back into the original equation: $\left(\frac{1}{36}\right)^n = 216$. Substituting $n = -\frac{3}{2}$, we get: $\left(\frac{1}{36}\right)^{-\frac{3}{2}} = 216$. Now, we need to simplify the left side of the equation and see if it equals $216$. Recall that $\frac{1}{36} = 6^{-2}$. So, the equation becomes: $(6^{-2})^{-\frac{3}{2}} = 216$. Applying the power of a power rule, $(a^b)^c = a^{bc}$, we multiply the exponents: $-2 \times -\frac{3}{2} = 3$. Thus, the left side simplifies to $6^3$. We know that $6^3 = 216$. Therefore, the equation becomes: $216 = 216$. Since the left side equals the right side, our solution $n = -\frac{3}{2}$ is correct. This verification step is a crucial part of the problem-solving process. It confirms that our solution is accurate and that we have not made any errors in our calculations. By verifying the solution, we gain confidence in our answer and ensure that we have correctly solved the problem. This practice is highly recommended in mathematics, as it helps to catch mistakes and reinforce understanding of the concepts involved. The verification process not only confirms the answer but also deepens our understanding of the relationship between the variables and the equation itself.

Conclusion

In conclusion, we successfully solved the exponential equation $\left(\frac{1}{36}\right)^n = 216$ by systematically applying the properties of exponents and algebraic manipulation. The key steps involved expressing both sides of the equation with a common base, simplifying the equation using the power of a power rule, equating the exponents, and solving for $n$. We found that $n = -\frac{3}{2}$. To ensure the accuracy of our solution, we verified it by substituting the value back into the original equation and confirming that both sides were equal. This problem exemplifies the importance of understanding and applying fundamental mathematical principles. By breaking down the problem into smaller, manageable steps, we were able to arrive at the correct solution. The process also highlights the interconnectedness of different mathematical concepts, such as exponents, algebra, and equation solving. Mastering these concepts is crucial for success in mathematics. The ability to solve exponential equations is a valuable skill that has applications in various fields, including science, engineering, and finance. By understanding the underlying principles and practicing problem-solving techniques, you can build confidence in your mathematical abilities and tackle more complex problems. Remember, mathematics is not just about finding the right answer; it's about understanding the process and the logic behind the solution.

Final Answer: The final answer is ${B) -\frac{3}{2}}$